From 7dabd38ce0da4ccb36673a1fb867f3e643ce30bd Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 17 Nov 2022 12:46:06 -0700 Subject: [PATCH 01/22] initial setup for our future editing and building --- Gemfile | 1 + _docs_v7/Convective-Schemes.md | 2 +- _docs_v7/Gradients-Limiters-Grant.md | 35 ++++++++++++++++++++++++++++ _docs_v7/Gradients-Limiters-Kal.md | 35 ++++++++++++++++++++++++++++ _docs_v7/Gradients-Limiters.md | 32 +++++++++++++++++++++++++ 5 files changed, 104 insertions(+), 1 deletion(-) create mode 100644 _docs_v7/Gradients-Limiters-Grant.md create mode 100644 _docs_v7/Gradients-Limiters-Kal.md create mode 100644 _docs_v7/Gradients-Limiters.md diff --git a/Gemfile b/Gemfile index e4bb3af0..9b5e72ad 100644 --- a/Gemfile +++ b/Gemfile @@ -2,4 +2,5 @@ source 'https://rubygems.org' group :jekyll_plugins do gem 'github-pages' gem 'jekyll-twitter-plugin' + gem 'webrick' end diff --git a/_docs_v7/Convective-Schemes.md b/_docs_v7/Convective-Schemes.md index 0195cbe8..2161153c 100755 --- a/_docs_v7/Convective-Schemes.md +++ b/_docs_v7/Convective-Schemes.md @@ -24,7 +24,7 @@ The options listed here do not apply to the high order DG solver. Convective schemes are used in the FVM discretization of convective fluxes through the faces of the dual-grid control volumes. They are selected via option `CONV_NUM_METHOD_FLOW` and fall under the two broad categories of central and upwind. Central schemes tend to be more robust whereas second order upwind schemes can be more accurate (i.e. less dissipative). -To achieve second order upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the "gradients and limiters" page for the MUSCL-related options. +To achieve second order upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the [Gradients and Limiters](/docs_v7/Gradients-Limiters) page for the MUSCL-related options. **Note:** MUSCL options have no effect on central schemes or on coarse multigrid levels in general. diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md new file mode 100644 index 00000000..009b642d --- /dev/null +++ b/_docs_v7/Gradients-Limiters-Grant.md @@ -0,0 +1,35 @@ +--- +title: Gradients and Limiters +permalink: /docs_v7/Gradients-Limiters-Grant/ +--- + +Write a brief summary once we are done with most of this. +Generally, the documenation is written with one sentence per line. + +--- + + +--- +## Grant version +? working on sections d,e,f + +## Basics of describing what options are available and providing some references for them +Do this before diving into the comparisons (which can be a lot more work) +Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. + +We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. + +## Why Slope Limiters are used in FVM +* TVD +* Monotonic +* 2nd order accuracy + +### subsection example +a subsection would go here + +## Mathematically describe limiters available to user in SU2 +Also discuss their properties. + +## Empirical comparison of the available limiters on a test problem +Flowfield colored by the limiter value. +Link to the Documentation on how to generate these \ No newline at end of file diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md new file mode 100644 index 00000000..e4690b47 --- /dev/null +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -0,0 +1,35 @@ +--- +title: Gradients and Limiters +permalink: /docs_v7/Gradients-Limiters-Kal/ +--- + +Write a brief summary once we are done with most of this. +Generally, the documenation is written with one sentence per line. + +--- + + +--- +## Kal version +? working on sections a,b,c + +## Basics of describing what options are available and providing some references for them +Do this before diving into the comparisons (which can be a lot more work) +Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. + +We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. + +## Why Slope Limiters are used in FVM +* TVD +* Monotonic +* 2nd order accuracy + +### subsection example +a subsection would go here + +## Mathematically describe limiters available to user in SU2 +Also discuss their properties. + +## Empirical comparison of the available limiters on a test problem +Flowfield colored by the limiter value. +Link to the Documentation on how to generate these \ No newline at end of file diff --git a/_docs_v7/Gradients-Limiters.md b/_docs_v7/Gradients-Limiters.md new file mode 100644 index 00000000..3a8ac858 --- /dev/null +++ b/_docs_v7/Gradients-Limiters.md @@ -0,0 +1,32 @@ +--- +title: Gradients and Limiters +permalink: /docs_v7/Gradients-Limiters/ +--- + +Write a brief summary once we are done with most of this. +Generally, the documenation is written with one sentence per line. + +--- + + +--- +## Basics of describing what options are available and providing some references for them +Do this before diving into the comparisons (which can be a lot more work) +Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. + +We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. + +## Why Slope Limiters are used in FVM +* TVD +* Monotonic +* 2nd order accuracy + +### subsection example +a subsection would go here + +## Mathematically describe limiters available to user in SU2 +Also discuss their properties. + +## Empirical comparison of the available limiters on a test problem +Flowfield colored by the limiter value. +Link to the Documentation on how to generate these \ No newline at end of file From 1d49d65e54a695523e43d5662f4db253fe5e02c1 Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Tue, 22 Nov 2022 01:08:54 -0700 Subject: [PATCH 02/22] "Updated what I'll be working on" --- _docs_v7/Gradients-Limiters-Grant.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md index 009b642d..8f9b4dd1 100644 --- a/_docs_v7/Gradients-Limiters-Grant.md +++ b/_docs_v7/Gradients-Limiters-Grant.md @@ -11,7 +11,11 @@ Generally, the documenation is written with one sentence per line. --- ## Grant version -? working on sections d,e,f +Grant will work on: +* "Basics of describing what options are available ..." +* "Mathematically describe limiters available in SU2" +* helping with any python scripts for postprocessing of empirical study +* general documenation logistics: building, markdown / html specifics, github pull requests ## Basics of describing what options are available and providing some references for them Do this before diving into the comparisons (which can be a lot more work) From be2d4bdbdc7991a59663e9e4f7ee678491203a96 Mon Sep 17 00:00:00 2001 From: kymonroe Date: Sun, 27 Nov 2022 16:34:47 -0500 Subject: [PATCH 03/22] Init commit for Kal's section. --- _docs_v7/Gradients-Limiters-Kal.md | 13 ++++++++++--- 1 file changed, 10 insertions(+), 3 deletions(-) diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index e4690b47..9d950b47 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -11,7 +11,11 @@ Generally, the documenation is written with one sentence per line. --- ## Kal version -? working on sections a,b,c +Kal will work on: +* "Why Slope Limiters are used in FVM" +* "Empirical comparison of the available limiters on a test problem" +* Help setting up and running any necessary SU2 simulations +* general documentation logistics, formatting, CURC navigation ## Basics of describing what options are available and providing some references for them Do this before diving into the comparisons (which can be a lot more work) @@ -20,6 +24,7 @@ Also we assume that the user will know the theory, and that they are just lookin We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. ## Why Slope Limiters are used in FVM +This section should describe the purpose slope limiters have in a FVM formulation. This should cover the concepts of: * TVD * Monotonic * 2nd order accuracy @@ -31,5 +36,7 @@ a subsection would go here Also discuss their properties. ## Empirical comparison of the available limiters on a test problem -Flowfield colored by the limiter value. -Link to the Documentation on how to generate these \ No newline at end of file +This section should describe in a general sense the characteristics of the limiters in SU2. These limiters include +* Minmod +* Venkatakrishnan +* Barth-Jespersen \ No newline at end of file From 396cb83dcaf8cbfe6cb339e982bf7bef6c61daa7 Mon Sep 17 00:00:00 2001 From: kymonroe Date: Tue, 29 Nov 2022 16:21:53 -0500 Subject: [PATCH 04/22] Added significant documentation to the 'Empirical comparison' section and the 'Why use slope limters?' section. Added figures showing (1) typical numerical oscillations that arise when resolving a shock (2) defining total variation (3) comparing the lax-wendroff scheme with the Barth-Jesperson, van-Albada, and Venkat limiters. --- _docs_v7/Gradients-Limiters-Kal.md | 67 +++++++++++++++++++++++------ docs_files/LW_example.png | Bin 0 -> 99627 bytes docs_files/TV_example.png | Bin 0 -> 36993 bytes docs_files/advection_example.png | Bin 0 -> 1328348 bytes 4 files changed, 55 insertions(+), 12 deletions(-) create mode 100644 docs_files/LW_example.png create mode 100644 docs_files/TV_example.png create mode 100644 docs_files/advection_example.png diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index 9d950b47..0eac68b1 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -23,20 +23,63 @@ Also we assume that the user will know the theory, and that they are just lookin We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. -## Why Slope Limiters are used in FVM -This section should describe the purpose slope limiters have in a FVM formulation. This should cover the concepts of: -* TVD -* Monotonic -* 2nd order accuracy +## Why are slope limiters used in a Finite Volume Method? +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as they provide high accuracy. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result on non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. -### subsection example -a subsection would go here + + +Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. + +SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. + +Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. + +### Total Variation and Total Variation Diminishing +We can first introduce the concept of **total variation** (TV) which is a measure of how oscillatory a solution is. In a discrete one dimensional setting, TV can be calculated as the following: + +$$ TV(u^n) = \sum_j |u^n_{j+1} - u^n_j| $$ + + + +Figure (2): A numerical scheme resulting in both high and low TV. + +A scheme can be said to be **total variation diminishing** (TVD) if + +$$ TV(u^{n+1}) \leq TV(u^n) $$ + +where for every successive timestep $$n$$, the total variation of the solution does not increase. + +A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. + +### Godunov's Theorem +The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to Godunov's Theorem. + +**Godunov's Theorem**: +1. A linear scheme is monotone if and only if it is total variation diminishing. +2. Linear total variation diminishing schemes are at most first-order accurate. + +The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. + +Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** + +The inclusion of a slope limiter into a TVD scheme accomplishes this idea. ## Mathematically describe limiters available to user in SU2 Also discuss their properties. -## Empirical comparison of the available limiters on a test problem -This section should describe in a general sense the characteristics of the limiters in SU2. These limiters include -* Minmod -* Venkatakrishnan -* Barth-Jespersen \ No newline at end of file +## Empirical comparison of limiters on a periodic advective domain +An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: + +$$ u_j^{n+1} = u_j^{n} - \sigma (u_j^{n} - u_{j-1}^{n}) - \frac{1}{2}\sigma(1-\sigma) \left[ \phi_{j+\frac{1}{2}}(u_{j+1}^{n} - u_j^{n}) - \phi_{j-\frac{1}{2}}(u_{j}^{n} - u_{j-1}^{n}) \right] $$ + +where $$\phi_{j+\frac{1}{2}}$$ is the scalar value of the limiter at the interface of cell $$u_j$$ and $$u_{j+1}$$. + + + +Figure (3): A one period advection (red) of an initial condition (black) using various schemes, with and without limiters. + +From the above example we note: +* The **Lax-Wendroff** scheme produces oscillations near sudden gradients due to dispersion errors. From Godunov's theorem this is expected as the scheme is second-order accurate and does not utilize a limiter. +* The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. +* The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. +* The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. 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Wow, such an elaborate caption](../../docs_files/LW_example.png){#fig:description} + + + ### subsection example a subsection would go here diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index 0eac68b1..b6095436 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -26,6 +26,10 @@ We may want to link to another place in the docs where they mention that limiter ## Why are slope limiters used in a Finite Volume Method? For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as they provide high accuracy. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result on non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. + + + + Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. @@ -58,6 +62,8 @@ The question of "How accurate can a TVD scheme be?" is still unanswered. For thi 1. A linear scheme is monotone if and only if it is total variation diminishing. 2. Linear total variation diminishing schemes are at most first-order accurate. + + The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** @@ -83,3 +89,5 @@ From the above example we note: * The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. * The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. * The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$k$$ that is flow dependent. + + From b73a4dde7618a183952d9cd66df9d5e13612272c Mon Sep 17 00:00:00 2001 From: kymonroe Date: Wed, 30 Nov 2022 16:02:08 -0500 Subject: [PATCH 06/22] Made changes to grammer, sentence structure according to comments by GN. --- _docs_v7/Gradients-Limiters-Kal.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index b6095436..df8b8678 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -24,11 +24,11 @@ Also we assume that the user will know the theory, and that they are just lookin We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. ## Why are slope limiters used in a Finite Volume Method? -For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as they provide high accuracy. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result on non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. - - - + + + @@ -56,13 +56,13 @@ where for every successive timestep $$n$$, the total variation of the solution d A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. ### Godunov's Theorem -The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to Godunov's Theorem. +The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to [Godunov's Theorem](https://en.wikipedia.org/wiki/Godunov%27s_theorem). **Godunov's Theorem**: 1. A linear scheme is monotone if and only if it is total variation diminishing. 2. Linear total variation diminishing schemes are at most first-order accurate. - + The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. @@ -90,4 +90,4 @@ From the above example we note: * The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. * The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$k$$ that is flow dependent. - + From 8f08f9f4476c4306b92744670f2b07d676da0f1d Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Wed, 30 Nov 2022 19:52:21 -0700 Subject: [PATCH 07/22] Options available for SU2 big chunk --- _docs_v7/Gradients-Limiters-Grant.md | 90 +++++++++++++++++++++++++--- 1 file changed, 83 insertions(+), 7 deletions(-) diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md index cea73539..fa5ccf48 100644 --- a/_docs_v7/Gradients-Limiters-Grant.md +++ b/_docs_v7/Gradients-Limiters-Grant.md @@ -21,7 +21,89 @@ Grant will work on: Do this before diving into the comparisons (which can be a lot more work) Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. -We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. + +The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use, and defaults to `VENKATAKRISHNAN` (see below). Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). +An example is shown here: https://su2code.github.io/tutorials/Laminar_Cylinder/. The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/ Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB` (also defaulting to `VENKATAKRISHNAN`), which is used for the turbulence equations, rather than for the flow equations. Using a limiter similarly requires `MUSCL_TURB = YES` The settings `MUSCL_ADJFLOW`, `MUSCL_ADJTURB`, `SLOPE_LIMITER_ADJFLOW`, and `SLOPE_LIMITER_ADJTURB` set the corresponding options for the respective adjoint equations. For species transport, the options `MUSCL_SPECIES` and `SLOPE_LIMITER_SPECIES` are used. + + + + + + +The `SLOPE_LIMITER_` options above may each be changed to use different limiters, as explained below. + + +All of the limiters depend on variables `proj`, `delta`, and `eps`/`eps2` (and maybe `dist` for `raisedSine` in `SHARP_EDGES`) + +Limiters + +| Type | Description | Notes | +| --- | --- | --- | +| `NONE` | No limiter | | +| `BARTH_JESPERSEN` | Barth-Jespersen | | +| `VENKATAKRISHNAN` | Venkatakrishnan | | +| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | +| `SHARP_EDGES` | Venkatakrishnan with sharp edge modification | | +| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | | + + +The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $0.05$, but its specific definition depends on the limiter being used. + +For the `VENKATAKRISHNAN` option, the `VENKAT_LIMITER_COEFF` parameter refers to $K$ in $\epsilon^2=\left(K\bar{\Delta} x\right)^3$, where $\bar{\Delta}$ is an average grid size. The $K$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. + + +[Wang](https://doi.org/10.2514/6.1996-2091) + + +After the number of iterations given by `LIMITER_ITER` (default $999999$), the value of the limiter will be frozen. + + + + + + + + + + + + + + + + + + + + + + + + + ## Why Slope Limiters are used in FVM * TVD @@ -29,12 +111,6 @@ We may want to link to another place in the docs where they mention that limiter * 2nd order accuracy -![Caption. Wow, such an elaborate caption](../../docs_files/LW_example.png){#fig:description} - - - ### subsection example a subsection would go here From b21564254a465f165ad947aa86c7c695cc206d7d Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 1 Dec 2022 03:13:17 -0700 Subject: [PATCH 08/22] Finished draft of describing SU2 options. --- _docs_v7/Gradients-Limiters-Grant.md | 103 ++++++++++++++++++--------- _docs_v7/Gradients-Limiters-Kal.md | 2 + 2 files changed, 71 insertions(+), 34 deletions(-) diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md index fa5ccf48..f217a9d3 100644 --- a/_docs_v7/Gradients-Limiters-Grant.md +++ b/_docs_v7/Gradients-Limiters-Grant.md @@ -18,17 +18,24 @@ Grant will work on: * general documenation logistics: building, markdown / html specifics, github pull requests ## Basics of describing what options are available and providing some references for them -Do this before diving into the comparisons (which can be a lot more work) -Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. + -The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use, and defaults to `VENKATAKRISHNAN` (see below). Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). -An example is shown here: https://su2code.github.io/tutorials/Laminar_Cylinder/. The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/ Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB` (also defaulting to `VENKATAKRISHNAN`), which is used for the turbulence equations, rather than for the flow equations. Using a limiter similarly requires `MUSCL_TURB = YES` The settings `MUSCL_ADJFLOW`, `MUSCL_ADJTURB`, `SLOPE_LIMITER_ADJFLOW`, and `SLOPE_LIMITER_ADJTURB` set the corresponding options for the respective adjoint equations. For species transport, the options `MUSCL_SPECIES` and `SLOPE_LIMITER_SPECIES` are used. +The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use. Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). +The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this. +The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations, rather than for the flow equations. +More possible applications of limiters are listed below. - + +Slope Limiter Fields + +| Configuration Field | Description | Notes | +| --- | --- | --- | +| `SLOPE_LIMITER_FLOW` | Flow equations | Need `MUSCL_FLOW = YES` | +| `SLOPE_LIMITER_TURB` | Turbulence equations | Need `MUSCL_TURB = YES` | +| `SLOPE_LIMITER_SPECIES` | Species evolution equations | Need `MUSCL_SPECIES = YES` | +| `SLOPE_LIMITER_ADJFLOW` | Adjoint flow equations | Need `MUSCL_ADJFLOW = YES` | +| `SLOPE_LIMITER_ADJTURB` | Adjoint turbulence equations | Need `MUSCL_ADJTURB = YES` | -The `SLOPE_LIMITER_` options above may each be changed to use different limiters, as explained below. +The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. -All of the limiters depend on variables `proj`, `delta`, and `eps`/`eps2` (and maybe `dist` for `raisedSine` in `SHARP_EDGES`) +Available Limiters -Limiters - -| Type | Description | Notes | -| --- | --- | --- | -| `NONE` | No limiter | | -| `BARTH_JESPERSEN` | Barth-Jespersen | | -| `VENKATAKRISHNAN` | Venkatakrishnan | | -| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | -| `SHARP_EDGES` | Venkatakrishnan with sharp edge modification | | -| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | | +| Type | Description | +| --- | --- | +| `NONE` | No limiter | +| `BARTH_JESPERSEN` | Barth-Jespersen | +| `VENKATAKRISHNAN` | Venkatakrishnan | +| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | +| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | +| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | +| `VAN_ALBADA_EDGE` | [^1] Van Albada (edge formulation) | -The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $0.05$, but its specific definition depends on the limiter being used. +[^1]: This limiter may or may not be implemented for certain solvers. It may also suffer from problems of not outputing limiter values. + -For the `VENKATAKRISHNAN` option, the `VENKAT_LIMITER_COEFF` parameter refers to $K$ in $\epsilon^2=\left(K\bar{\Delta} x\right)^3$, where $\bar{\Delta}$ is an average grid size. The $K$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. +The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. +This is different than the small constant used to prevent division by zero, which is used for all limiters. -[Wang](https://doi.org/10.2514/6.1996-2091) - - -After the number of iterations given by `LIMITER_ITER` (default $999999$), the value of the limiter will be frozen. - - - - - +For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. +The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). +Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. +This value depends on both the mesh and the flow variable. + + + +Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`. +This parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. +When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. +Note that this global operation may incur extra time costs due to communication between MPI threads. +Based on the original work by [Wang](https://doi.org/10.2514/6.1996-2091) introducing this limiter suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. +The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**. +A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**. +For both the `SHARP_EDGES` and `WALL_DISTANCE` limiters, the influence of the geometric modifier is controlled with `ADJ_SHARP_LIMITER_COEFF` which defaults to 3.0. +Increasing this parameter will decrease the value of the limiter and thus make the field more diffusive and less oscillatory near the feature (sharp edge or wall). +In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is described by the parameter `REF_SHARP_EDGES` (defaults to 3.0). Increasing this will make more edges qualify as "sharp". +Other than the addition of this geometric factor, these limiters are the same as the `VENKATAKRISHNAN` limiter and should also use `VENKAT_LIMITER_COEFF` (given by $$K$$ below). + +Specifically, given the distance to the feature, $$d_{\text{feature}}$$, an intermediate measure of the distance, $$d$$, is calculated. The parameter $$c$$ is set by `ADJ_SHARP_LIMITER_COEFF`. +$$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$ +Then, the geometric factor is given by +$$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$ +Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$. + + +After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen. +The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen in the discrete adjoint formulation (default is `NO`). + + - - ## Why Slope Limiters are used in FVM * TVD * Monotonic @@ -117,6 +144,14 @@ a subsection would go here ## Mathematically describe limiters available to user in SU2 Also discuss their properties. + + + + ## Empirical comparison of the available limiters on a test problem Flowfield colored by the limiter value. Link to the Documentation on how to generate these \ No newline at end of file diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index df8b8678..0d7ae948 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -29,6 +29,8 @@ For many studying compressible flow or high-speed aerodynamics, the formation of + From 15976f8b27b3260b7dc8bfe0cb1593fca14ac03b Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 1 Dec 2022 05:52:31 -0700 Subject: [PATCH 09/22] premerge backup --- _docs_v7/Gradients-Limiters-Grant.md | 6 ++++-- _docs_v7/Gradients-Limiters-Kal.md | 2 +- 2 files changed, 5 insertions(+), 3 deletions(-) diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md index f217a9d3..50b8618b 100644 --- a/_docs_v7/Gradients-Limiters-Grant.md +++ b/_docs_v7/Gradients-Limiters-Grant.md @@ -17,6 +17,7 @@ Grant will work on: * helping with any python scripts for postprocessing of empirical study * general documenation logistics: building, markdown / html specifics, github pull requests + ## Basics of describing what options are available and providing some references for them @@ -63,6 +64,7 @@ Available Limiters | `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | | `VAN_ALBADA_EDGE` | [^1] Van Albada (edge formulation) | + [^1]: This limiter may or may not be implemented for certain solvers. It may also suffer from problems of not outputing limiter values. @@ -146,10 +148,10 @@ Also discuss their properties. + ## Empirical comparison of the available limiters on a test problem diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index 0d7ae948..1dd9a065 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -30,7 +30,7 @@ For many studying compressible flow or high-speed aerodynamics, the formation of +definitely "second-order" when used as an adjective, but not sure if "second-order" when used as a noun.--> From 7674883865a7c58182ca7e15b8558ea3f74816de Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 1 Dec 2022 06:01:10 -0700 Subject: [PATCH 10/22] post merge --- _docs_v7/Gradients-Limiters-Grant.md | 159 -------------------- _docs_v7/Gradients-Limiters-Kal.md | 3 +- _docs_v7/Gradients-Limiters.md | 211 +++++++++++++++++++++++++-- 3 files changed, 201 insertions(+), 172 deletions(-) delete mode 100644 _docs_v7/Gradients-Limiters-Grant.md diff --git a/_docs_v7/Gradients-Limiters-Grant.md b/_docs_v7/Gradients-Limiters-Grant.md deleted file mode 100644 index 50b8618b..00000000 --- a/_docs_v7/Gradients-Limiters-Grant.md +++ /dev/null @@ -1,159 +0,0 @@ ---- -title: Gradients and Limiters -permalink: /docs_v7/Gradients-Limiters-Grant/ ---- - -Write a brief summary once we are done with most of this. -Generally, the documenation is written with one sentence per line. - ---- - - ---- -## Grant version -Grant will work on: -* "Basics of describing what options are available ..." -* "Mathematically describe limiters available in SU2" -* helping with any python scripts for postprocessing of empirical study -* general documenation logistics: building, markdown / html specifics, github pull requests - - -## Basics of describing what options are available and providing some references for them - - - -The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use. Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). -The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this. -The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations, rather than for the flow equations. -More possible applications of limiters are listed below. - - -Slope Limiter Fields - -| Configuration Field | Description | Notes | -| --- | --- | --- | -| `SLOPE_LIMITER_FLOW` | Flow equations | Need `MUSCL_FLOW = YES` | -| `SLOPE_LIMITER_TURB` | Turbulence equations | Need `MUSCL_TURB = YES` | -| `SLOPE_LIMITER_SPECIES` | Species evolution equations | Need `MUSCL_SPECIES = YES` | -| `SLOPE_LIMITER_ADJFLOW` | Adjoint flow equations | Need `MUSCL_ADJFLOW = YES` | -| `SLOPE_LIMITER_ADJTURB` | Adjoint turbulence equations | Need `MUSCL_ADJTURB = YES` | - - - - -The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. - - -Available Limiters - -| Type | Description | -| --- | --- | -| `NONE` | No limiter | -| `BARTH_JESPERSEN` | Barth-Jespersen | -| `VENKATAKRISHNAN` | Venkatakrishnan | -| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | -| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | -| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | -| `VAN_ALBADA_EDGE` | [^1] Van Albada (edge formulation) | - - - -[^1]: This limiter may or may not be implemented for certain solvers. It may also suffer from problems of not outputing limiter values. - - - -The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. -This is different than the small constant used to prevent division by zero, which is used for all limiters. - -For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. -The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). -Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. -This value depends on both the mesh and the flow variable. - - - - -Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`. -This parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. - -When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. -Note that this global operation may incur extra time costs due to communication between MPI threads. -Based on the original work by [Wang](https://doi.org/10.2514/6.1996-2091) introducing this limiter suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. - -The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**. -A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**. - -For both the `SHARP_EDGES` and `WALL_DISTANCE` limiters, the influence of the geometric modifier is controlled with `ADJ_SHARP_LIMITER_COEFF` which defaults to 3.0. -Increasing this parameter will decrease the value of the limiter and thus make the field more diffusive and less oscillatory near the feature (sharp edge or wall). - -In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is described by the parameter `REF_SHARP_EDGES` (defaults to 3.0). Increasing this will make more edges qualify as "sharp". -Other than the addition of this geometric factor, these limiters are the same as the `VENKATAKRISHNAN` limiter and should also use `VENKAT_LIMITER_COEFF` (given by $$K$$ below). - - - -Specifically, given the distance to the feature, $$d_{\text{feature}}$$, an intermediate measure of the distance, $$d$$, is calculated. The parameter $$c$$ is set by `ADJ_SHARP_LIMITER_COEFF`. - -$$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$ - -Then, the geometric factor is given by - -$$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$ - -Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$. - - - - - -After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen. - - -The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen in the discrete adjoint formulation (default is `NO`). - - - - - -## Why Slope Limiters are used in FVM -* TVD -* Monotonic -* 2nd order accuracy - - -### subsection example -a subsection would go here - -## Mathematically describe limiters available to user in SU2 -Also discuss their properties. - - - - - - -## Empirical comparison of the available limiters on a test problem -Flowfield colored by the limiter value. -Link to the Documentation on how to generate these \ No newline at end of file diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md index 1dd9a065..c3ccd3a9 100644 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ b/_docs_v7/Gradients-Limiters-Kal.md @@ -30,7 +30,8 @@ For many studying compressible flow or high-speed aerodynamics, the formation of +definitely "second-order" when used as an adjective, but not sure if "second-order" when used as +--> diff --git a/_docs_v7/Gradients-Limiters.md b/_docs_v7/Gradients-Limiters.md index 3a8ac858..785efa96 100644 --- a/_docs_v7/Gradients-Limiters.md +++ b/_docs_v7/Gradients-Limiters.md @@ -10,23 +10,210 @@ Generally, the documenation is written with one sentence per line. --- + + + ## Basics of describing what options are available and providing some references for them -Do this before diving into the comparisons (which can be a lot more work) -Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. + + + +The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use. Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). +The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this. +The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations, rather than for the flow equations. +More possible applications of limiters are listed below. + + +Slope Limiter Fields + +| Configuration Field | Description | Notes | +| --- | --- | --- | +| `SLOPE_LIMITER_FLOW` | Flow equations | Need `MUSCL_FLOW = YES` | +| `SLOPE_LIMITER_TURB` | Turbulence equations | Need `MUSCL_TURB = YES` | +| `SLOPE_LIMITER_SPECIES` | Species evolution equations | Need `MUSCL_SPECIES = YES` | +| `SLOPE_LIMITER_ADJFLOW` | Adjoint flow equations | Need `MUSCL_ADJFLOW = YES` | +| `SLOPE_LIMITER_ADJTURB` | Adjoint turbulence equations | Need `MUSCL_ADJTURB = YES` | + + + + +The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. + + +Available Limiters + +| Type | Description | Note | +| --- | --- | --- | +| `NONE` | No limiter | | +| `BARTH_JESPERSEN` | Barth-Jespersen | | +| `VENKATAKRISHNAN` | Venkatakrishnan | | +| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | +| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | | +| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | | +| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter may or may not be implemented for certain solvers [^1] | + +[^1]: It may also suffer from problems of not outputing limiter values. + + + +The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. +This is different than the small constant used to prevent division by zero, which is used for all limiters. + +For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. +The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). +Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. +This value depends on both the mesh and the flow variable. + + + + +Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`. +This parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. + +When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. +Note that this global operation may incur extra time costs due to communication between MPI threads. +Based on the original work by [Wang](https://doi.org/10.2514/6.1996-2091) introducing this limiter suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. + +The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**. +A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**. + +For both the `SHARP_EDGES` and `WALL_DISTANCE` limiters, the influence of the geometric modifier is controlled with `ADJ_SHARP_LIMITER_COEFF` which defaults to 3.0. +Increasing this parameter will decrease the value of the limiter and thus make the field more diffusive and less oscillatory near the feature (sharp edge or wall). + +In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is described by the parameter `REF_SHARP_EDGES` (defaults to 3.0). Increasing this will make more edges qualify as "sharp". +Other than the addition of this geometric factor, these limiters are the same as the `VENKATAKRISHNAN` limiter and should also use `VENKAT_LIMITER_COEFF` (given by $$K$$ below). + + + +Specifically, given the distance to the feature, $$d_{\text{feature}}$$, an intermediate measure of the distance, $$d$$, is calculated. The parameter $$c$$ is set by `ADJ_SHARP_LIMITER_COEFF`. + +$$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$ + +Then, the geometric factor is given by + +$$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$ + +Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$. + + + + + +After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen. + + +The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen in the discrete adjoint formulation (default is `NO`). + + + + + +## Why are slope limiters used in a Finite Volume Method? +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. + + + + + + + + +Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. + +SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. + +Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. + +### Total Variation and Total Variation Diminishing +We can first introduce the concept of **total variation** (TV) which is a measure of how oscillatory a solution is. In a discrete one dimensional setting, TV can be calculated as the following: + +$$ TV(u^n) = \sum_j |u^n_{j+1} - u^n_j| $$ + + + +Figure (2): A numerical scheme resulting in both high and low TV. + +A scheme can be said to be **total variation diminishing** (TVD) if + +$$ TV(u^{n+1}) \leq TV(u^n) $$ + +where for every successive timestep $$n$$, the total variation of the solution does not increase. + +A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. + +### Godunov's Theorem +The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to [Godunov's Theorem](https://en.wikipedia.org/wiki/Godunov%27s_theorem). + +**Godunov's Theorem**: +1. A linear scheme is monotone if and only if it is total variation diminishing. +2. Linear total variation diminishing schemes are at most first-order accurate. + + + +The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. + +Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** + +The inclusion of a slope limiter into a TVD scheme accomplishes this idea. -We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. -## Why Slope Limiters are used in FVM -* TVD -* Monotonic -* 2nd order accuracy -### subsection example -a subsection would go here ## Mathematically describe limiters available to user in SU2 Also discuss their properties. -## Empirical comparison of the available limiters on a test problem -Flowfield colored by the limiter value. -Link to the Documentation on how to generate these \ No newline at end of file + + + + + +## Empirical comparison of limiters on a periodic advective domain +An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: + +$$ u_j^{n+1} = u_j^{n} - \sigma (u_j^{n} - u_{j-1}^{n}) - \frac{1}{2}\sigma(1-\sigma) \left[ \phi_{j+\frac{1}{2}}(u_{j+1}^{n} - u_j^{n}) - \phi_{j-\frac{1}{2}}(u_{j}^{n} - u_{j-1}^{n}) \right] $$ + +where $$\phi_{j+\frac{1}{2}}$$ is the scalar value of the limiter at the interface of cell $$u_j$$ and $$u_{j+1}$$. + + + +Figure (3): A one period advection (red) of an initial condition (black) using various schemes, with and without limiters. + +From the above example we note: +* The **Lax-Wendroff** scheme produces oscillations near sudden gradients due to dispersion errors. From Godunov's theorem this is expected as the scheme is second-order accurate and does not utilize a limiter. +* The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. +* The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. +* The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$k$$ that is flow dependent. + + From 18397a5df9b927b395f698ee3063175bb74c63c9 Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 1 Dec 2022 06:45:00 -0700 Subject: [PATCH 11/22] More changes and changing subtitles --- _docs_v7/Gradients-Limiters.md | 71 ++++++++++++++++++++-------------- 1 file changed, 42 insertions(+), 29 deletions(-) diff --git a/_docs_v7/Gradients-Limiters.md b/_docs_v7/Gradients-Limiters.md index 785efa96..5dd44cb7 100644 --- a/_docs_v7/Gradients-Limiters.md +++ b/_docs_v7/Gradients-Limiters.md @@ -7,7 +7,8 @@ Write a brief summary once we are done with most of this. Generally, the documenation is written with one sentence per line. --- - + +- [Available Limiter Options](#available-limiter-options) --- -## Basics of describing what options are available and providing some references for them +## Available Limiter Options @@ -33,8 +34,8 @@ The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) sh The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations, rather than for the flow equations. More possible applications of limiters are listed below. - -Slope Limiter Fields + +### Slope Limiter Fields | Configuration Field | Description | Notes | | --- | --- | --- | @@ -49,61 +50,72 @@ Slope Limiter Fields CLimiterDetails.hpp: computeLimiters.hpp uses computeLimiters_impl.hpp and really just sets the limiter computeLimiters_impl.hpp passes computation of field values (cell averages), gradients, and to the limiter - also check out -computeGradientsGreenGauss.hpp - +computeGradientsGreenGauss.hpp (and the least squares one) --> The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. + +**Note:** the Discontinuous-Galerkin methods (DG) / Higher-order methods (HOM) do not use limiters. + -Available Limiters +### Available Limiters -| Type | Description | Note | +| Type | Description | Notes | | --- | --- | --- | | `NONE` | No limiter | | | `BARTH_JESPERSEN` | Barth-Jespersen | | | `VENKATAKRISHNAN` | Venkatakrishnan | | | `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | -| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | | -| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | | -| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter may or may not be implemented for certain solvers [^1] | +| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | This limiter should not be used for flow solvers | +| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | This limiter should not be used for flow solvers | +| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter may or may not be implemented for certain solvers, and it may also suffer from other issues, such as not outputing limiter values | -[^1]: It may also suffer from problems of not outputing limiter values. - + + + +The default limiter is `VENKATAKRISHNAN`. + +### Limiter Parameters and Further Details The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. -This is different than the small constant used to prevent division by zero, which is used for all limiters. +This is different than the small constant used to prevent division by zero, which is used in all these limiters. + For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). -Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence. -This value depends on both the mesh and the flow variable. +Thus, a large value will approach the case of using no limiter with undamped oscillations, while too small of a value will slow the convergence and add extra diffusion. +This value depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. - + + + + Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`. -This parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. +The `REF_ELEM_LENGTH` parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. Note that this global operation may incur extra time costs due to communication between MPI threads. -Based on the original work by [Wang](https://doi.org/10.2514/6.1996-2091) introducing this limiter suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. +The original work by [Wang](https://doi.org/10.2514/6.1996-2091) suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**. A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**. For both the `SHARP_EDGES` and `WALL_DISTANCE` limiters, the influence of the geometric modifier is controlled with `ADJ_SHARP_LIMITER_COEFF` which defaults to 3.0. +**Note:** these limiters should not be used for flow solvers. + + Increasing this parameter will decrease the value of the limiter and thus make the field more diffusive and less oscillatory near the feature (sharp edge or wall). In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is described by the parameter `REF_SHARP_EDGES` (defaults to 3.0). Increasing this will make more edges qualify as "sharp". Other than the addition of this geometric factor, these limiters are the same as the `VENKATAKRISHNAN` limiter and should also use `VENKAT_LIMITER_COEFF` (given by $$K$$ below). - - Specifically, given the distance to the feature, $$d_{\text{feature}}$$, an intermediate measure of the distance, $$d$$, is calculated. The parameter $$c$$ is set by `ADJ_SHARP_LIMITER_COEFF`. + $$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$ @@ -126,7 +138,7 @@ The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen -## Why are slope limiters used in a Finite Volume Method? +## Theory: Purpose of Slope Limiters For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. @@ -190,11 +202,12 @@ The inclusion of a slope limiter into a TVD scheme accomplishes this idea. + + + -## Mathematically describe limiters available to user in SU2 -Also discuss their properties. - - + + @@ -214,6 +227,6 @@ From the above example we note: * The **Lax-Wendroff** scheme produces oscillations near sudden gradients due to dispersion errors. From Godunov's theorem this is expected as the scheme is second-order accurate and does not utilize a limiter. * The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. * The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. -* The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$k$$ that is flow dependent. +* The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$K$$ that is flow dependent. From b6eeaeb51e8e23f01c92097cb51bb31640d146b2 Mon Sep 17 00:00:00 2001 From: kymonroe Date: Thu, 1 Dec 2022 20:37:04 -0700 Subject: [PATCH 12/22] made changes to order. Now: Theory and backgorund --> available limiter options --> empirical comparison. Added introduction and changed grammer in a few places --- _docs_v7/Gradients-Limiters.md | 141 ++++++++++++++++++--------------- 1 file changed, 75 insertions(+), 66 deletions(-) diff --git a/_docs_v7/Gradients-Limiters.md b/_docs_v7/Gradients-Limiters.md index 5dd44cb7..8a114acf 100644 --- a/_docs_v7/Gradients-Limiters.md +++ b/_docs_v7/Gradients-Limiters.md @@ -3,12 +3,19 @@ title: Gradients and Limiters permalink: /docs_v7/Gradients-Limiters/ --- -Write a brief summary once we are done with most of this. -Generally, the documenation is written with one sentence per line. +This page lists the limiters available in SU2 and their associated options, it is not meant as a detailed theory guide but a brief review of the governing mathematics is presented. +The options listed here do not apply to the high order DG solver. --- +- [Theory: An introduction to slope limiters](#theory-an-introduction-to-slope-limiters) + - [Total Variation and Total Variation Diminishing](#total-variation-and-total-variation-diminishing) + - [Godunov's Theorem](#godunovs-theorem) - [Available Limiter Options](#available-limiter-options) + - [Slope Limiter Fields](#slope-limiter-fields) + - [Available Limiters](#available-limiters) + - [Limiter Parameters and Further Details](#limiter-parameters-and-further-details) +- [Empirical comparison of limiters on a periodic advective domain](#empirical-comparison-of-limiters-on-a-periodic-advective-domain) --- +## Theory: An introduction to slope limiters +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. + + + + + + + + +Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. + +SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. + +Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. + +### Total Variation and Total Variation Diminishing +We can first introduce the concept of **total variation** (TV) which is a measure of how oscillatory a solution is. In a discrete one dimensional setting, TV can be calculated as the following: + +$$ TV(u^n) = \sum_j |u^n_{j+1} - u^n_j| $$ + + + +Figure (2): A numerical scheme resulting in both high and low TV. + +A scheme can be said to be **total variation diminishing** (TVD) if + +$$ TV(u^{n+1}) \leq TV(u^n) $$ + +where for every successive timestep $$n$$, the total variation of the solution does not increase. + +A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. + +### Godunov's Theorem +The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to [Godunov's Theorem](https://en.wikipedia.org/wiki/Godunov%27s_theorem). + +**Godunov's Theorem**: +1. A linear scheme is monotone if and only if it is total variation diminishing. +2. Linear total variation diminishing schemes are at most first-order accurate. + + + +The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. + +Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** + +The inclusion of a slope limiter into a TVD scheme accomplishes this idea. + + + + + + + + + + + + + + ## Available Limiter Options @@ -150,70 +220,8 @@ SU2_MPI::Error("Unknown limiter type.", CURRENT_FUNCTION); --> -## Theory: Purpose of Slope Limiters -For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. - - - - - - - - -Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. - -SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. - -Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. - -### Total Variation and Total Variation Diminishing -We can first introduce the concept of **total variation** (TV) which is a measure of how oscillatory a solution is. In a discrete one dimensional setting, TV can be calculated as the following: - -$$ TV(u^n) = \sum_j |u^n_{j+1} - u^n_j| $$ - - - -Figure (2): A numerical scheme resulting in both high and low TV. - -A scheme can be said to be **total variation diminishing** (TVD) if - -$$ TV(u^{n+1}) \leq TV(u^n) $$ - -where for every successive timestep $$n$$, the total variation of the solution does not increase. - -A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. - -### Godunov's Theorem -The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to [Godunov's Theorem](https://en.wikipedia.org/wiki/Godunov%27s_theorem). - -**Godunov's Theorem**: -1. A linear scheme is monotone if and only if it is total variation diminishing. -2. Linear total variation diminishing schemes are at most first-order accurate. - - - -The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. - -Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** - -The inclusion of a slope limiter into a TVD scheme accomplishes this idea. - - - - - - - - - - - - - ## Empirical comparison of limiters on a periodic advective domain -An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: +An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the primary limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: $$ u_j^{n+1} = u_j^{n} - \sigma (u_j^{n} - u_{j-1}^{n}) - \frac{1}{2}\sigma(1-\sigma) \left[ \phi_{j+\frac{1}{2}}(u_{j+1}^{n} - u_j^{n}) - \phi_{j-\frac{1}{2}}(u_{j}^{n} - u_{j-1}^{n}) \right] $$ @@ -229,4 +237,5 @@ From the above example we note: * The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. * The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$K$$ that is flow dependent. - + From fe8ce388300e74769bfdc47c9d81d8bce305f538 Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Thu, 1 Dec 2022 09:41:01 -0700 Subject: [PATCH 13/22] fixing messup of editing before pulling. --- _docs_v7/Convective-Schemes.md | 2 +- _docs_v7/Gradients-Limiters-Kal.md | 96 ------------------- ...=> Slope-Limiters-and-Shock-Resolution.md} | 52 ++-------- 3 files changed, 9 insertions(+), 141 deletions(-) delete mode 100644 _docs_v7/Gradients-Limiters-Kal.md rename _docs_v7/{Gradients-Limiters.md => Slope-Limiters-and-Shock-Resolution.md} (86%) diff --git a/_docs_v7/Convective-Schemes.md b/_docs_v7/Convective-Schemes.md index 2161153c..95a81960 100755 --- a/_docs_v7/Convective-Schemes.md +++ b/_docs_v7/Convective-Schemes.md @@ -24,7 +24,7 @@ The options listed here do not apply to the high order DG solver. Convective schemes are used in the FVM discretization of convective fluxes through the faces of the dual-grid control volumes. They are selected via option `CONV_NUM_METHOD_FLOW` and fall under the two broad categories of central and upwind. Central schemes tend to be more robust whereas second order upwind schemes can be more accurate (i.e. less dissipative). -To achieve second order upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the [Gradients and Limiters](/docs_v7/Gradients-Limiters) page for the MUSCL-related options. +To achieve second order upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the [Slope Limiters and Shock Resolution](/docs_v7/Slope-Limiters-and-Shock-Resolution) page for the MUSCL-related options. **Note:** MUSCL options have no effect on central schemes or on coarse multigrid levels in general. diff --git a/_docs_v7/Gradients-Limiters-Kal.md b/_docs_v7/Gradients-Limiters-Kal.md deleted file mode 100644 index c3ccd3a9..00000000 --- a/_docs_v7/Gradients-Limiters-Kal.md +++ /dev/null @@ -1,96 +0,0 @@ ---- -title: Gradients and Limiters -permalink: /docs_v7/Gradients-Limiters-Kal/ ---- - -Write a brief summary once we are done with most of this. -Generally, the documenation is written with one sentence per line. - ---- - - ---- -## Kal version -Kal will work on: -* "Why Slope Limiters are used in FVM" -* "Empirical comparison of the available limiters on a test problem" -* Help setting up and running any necessary SU2 simulations -* general documentation logistics, formatting, CURC navigation - -## Basics of describing what options are available and providing some references for them -Do this before diving into the comparisons (which can be a lot more work) -Also we assume that the user will know the theory, and that they are just looking for the limiters that are available in SU2 first. - -We may want to link to another place in the docs where they mention that limiters can be activated after a specific number of iterations. - -## Why are slope limiters used in a Finite Volume Method? -For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. - - - - - - - - -Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. - -SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. - -Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. - -### Total Variation and Total Variation Diminishing -We can first introduce the concept of **total variation** (TV) which is a measure of how oscillatory a solution is. In a discrete one dimensional setting, TV can be calculated as the following: - -$$ TV(u^n) = \sum_j |u^n_{j+1} - u^n_j| $$ - - - -Figure (2): A numerical scheme resulting in both high and low TV. - -A scheme can be said to be **total variation diminishing** (TVD) if - -$$ TV(u^{n+1}) \leq TV(u^n) $$ - -where for every successive timestep $$n$$, the total variation of the solution does not increase. - -A favorable property of TVD schemes is that they are **monotonicity preserving**. This means they do not introduce new extrema into the solution and local minimum (maximum) are non-decreasing (increasing) in time. These are both desirable qualities for correctly resolving a shock and ensuring the solution is physical. - -### Godunov's Theorem -The question of "How accurate can a TVD scheme be?" is still unanswered. For this, we turn to [Godunov's Theorem](https://en.wikipedia.org/wiki/Godunov%27s_theorem). - -**Godunov's Theorem**: -1. A linear scheme is monotone if and only if it is total variation diminishing. -2. Linear total variation diminishing schemes are at most first-order accurate. - - - -The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. - -Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** - -The inclusion of a slope limiter into a TVD scheme accomplishes this idea. - -## Mathematically describe limiters available to user in SU2 -Also discuss their properties. - -## Empirical comparison of limiters on a periodic advective domain -An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: - -$$ u_j^{n+1} = u_j^{n} - \sigma (u_j^{n} - u_{j-1}^{n}) - \frac{1}{2}\sigma(1-\sigma) \left[ \phi_{j+\frac{1}{2}}(u_{j+1}^{n} - u_j^{n}) - \phi_{j-\frac{1}{2}}(u_{j}^{n} - u_{j-1}^{n}) \right] $$ - -where $$\phi_{j+\frac{1}{2}}$$ is the scalar value of the limiter at the interface of cell $$u_j$$ and $$u_{j+1}$$. - - - -Figure (3): A one period advection (red) of an initial condition (black) using various schemes, with and without limiters. - -From the above example we note: -* The **Lax-Wendroff** scheme produces oscillations near sudden gradients due to dispersion errors. From Godunov's theorem this is expected as the scheme is second-order accurate and does not utilize a limiter. -* The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. -* The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. -* The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$k$$ that is flow dependent. - - diff --git a/_docs_v7/Gradients-Limiters.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md similarity index 86% rename from _docs_v7/Gradients-Limiters.md rename to _docs_v7/Slope-Limiters-and-Shock-Resolution.md index 8a114acf..5a378891 100644 --- a/_docs_v7/Gradients-Limiters.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -1,13 +1,13 @@ --- -title: Gradients and Limiters -permalink: /docs_v7/Gradients-Limiters/ +title: Slope Limiters and Shock Resolution +permalink: /docs_v7/Slope-Limiters-and-Shock-Resolution/ --- This page lists the limiters available in SU2 and their associated options, it is not meant as a detailed theory guide but a brief review of the governing mathematics is presented. The options listed here do not apply to the high order DG solver. --- - + - [Theory: An introduction to slope limiters](#theory-an-introduction-to-slope-limiters) - [Total Variation and Total Variation Diminishing](#total-variation-and-total-variation-diminishing) - [Godunov's Theorem](#godunovs-theorem) @@ -18,30 +18,12 @@ The options listed here do not apply to the high order DG solver. - [Empirical comparison of limiters on a periodic advective domain](#empirical-comparison-of-limiters-on-a-periodic-advective-domain) --- - ## Theory: An introduction to slope limiters For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. - - - - - Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. @@ -74,8 +56,6 @@ The question of "How accurate can a TVD scheme be?" is still unanswered. For thi 1. A linear scheme is monotone if and only if it is total variation diminishing. 2. Linear total variation diminishing schemes are at most first-order accurate. - - The first statement is simple, stating that for linear schemes, the characteristic of being monotone and TVD is equivalent. The second statement is more interesting. It states that if we want to construct a linear TVD (monotone) scheme, the best we can be possibly hope for is first-order accuracy. Recall that the original motivation for a slope limiter was to prevent the formation of oscillations in the solution. In the section above, we noted that TVD schemes are monotonicity preserving (a favorable property in resolving a shock). However, through Godunov's theorem, we note that if we also want high-order accuracy, **our TVD discretization MUST be nonlinear** @@ -84,20 +64,11 @@ The inclusion of a slope limiter into a TVD scheme accomplishes this idea. - - + - - - - - - ## Available Limiter Options - - The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use. Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this. @@ -125,9 +96,9 @@ computeGradientsGreenGauss.hpp (and the least squares one) --> The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. - + **Note:** the Discontinuous-Galerkin methods (DG) / Higher-order methods (HOM) do not use limiters. - + ### Available Limiters @@ -140,9 +111,8 @@ The `SLOPE_LIMITER_` options above may each be changed to use different limiters | `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | | `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | This limiter should not be used for flow solvers | | `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | This limiter should not be used for flow solvers | -| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter may or may not be implemented for certain solvers, and it may also suffer from other issues, such as not outputing limiter values | +| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter has been observed to not output limiter values when called using VOLUME_OUTPUT | - @@ -185,7 +155,7 @@ In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is Other than the addition of this geometric factor, these limiters are the same as the `VENKATAKRISHNAN` limiter and should also use `VENKAT_LIMITER_COEFF` (given by $$K$$ below). Specifically, given the distance to the feature, $$d_{\text{feature}}$$, an intermediate measure of the distance, $$d$$, is calculated. The parameter $$c$$ is set by `ADJ_SHARP_LIMITER_COEFF`. - + $$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$ @@ -196,9 +166,6 @@ $$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$ Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$. - - - After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen. @@ -236,6 +203,3 @@ From the above example we note: * The **Barth-Jespersen** limiter performs well for most of the waveforms. However, the Barth-Jespersen limtier is known to be compressive and will turn smooth waves into square waves. This is best seen with the value discontinuity on the very left. * The **Van-Albada** limiter also performs well. It is slightly more diffusive than Barth-Jespersen but has robust convergence properties. * The **Venkatakrishnan** limiter is similar to the Barth-Jespersen and has significantly improved convergence properties. However, it is more diffusive and does require a user-specified parameter $$K$$ that is flow dependent. - - From 54c6d161102bd9ddbafd530b65d5c792c58f2678 Mon Sep 17 00:00:00 2001 From: kymonroe Date: Thu, 1 Dec 2022 21:56:00 -0700 Subject: [PATCH 14/22] fixed small latex error --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 5a378891..8fe066b2 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -163,7 +163,7 @@ Then, the geometric factor is given by $$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$ -Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$. +Note that the geometric factor is nonnegative and nondecreasing in $$d_{feature}$$. After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen. From 89471a949cb4251b98e3725c9baaed537ca4efb5 Mon Sep 17 00:00:00 2001 From: Grant Norman <36522261+gnorman7@users.noreply.github.com> Date: Sun, 4 Dec 2022 13:37:45 -0700 Subject: [PATCH 15/22] Update _docs_v7/Convective-Schemes.md Co-authored-by: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> --- _docs_v7/Convective-Schemes.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Convective-Schemes.md b/_docs_v7/Convective-Schemes.md index 95a81960..6c7f446b 100755 --- a/_docs_v7/Convective-Schemes.md +++ b/_docs_v7/Convective-Schemes.md @@ -24,7 +24,7 @@ The options listed here do not apply to the high order DG solver. Convective schemes are used in the FVM discretization of convective fluxes through the faces of the dual-grid control volumes. They are selected via option `CONV_NUM_METHOD_FLOW` and fall under the two broad categories of central and upwind. Central schemes tend to be more robust whereas second order upwind schemes can be more accurate (i.e. less dissipative). -To achieve second order upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the [Slope Limiters and Shock Resolution](/docs_v7/Slope-Limiters-and-Shock-Resolution) page for the MUSCL-related options. +To achieve second-order in space, upwind schemes need to be used with MUSCL reconstruction (`MUSCL_FLOW = YES`), see the [Slope Limiters and Shock Resolution](/docs_v7/Slope-Limiters-and-Shock-Resolution) page for the MUSCL-related options. **Note:** MUSCL options have no effect on central schemes or on coarse multigrid levels in general. From acefa7cdae343f6c2a14064d7646b53f5f0301ab Mon Sep 17 00:00:00 2001 From: Grant Norman <36522261+gnorman7@users.noreply.github.com> Date: Sun, 4 Dec 2022 13:48:42 -0700 Subject: [PATCH 16/22] Update _docs_v7/Slope-Limiters-and-Shock-Resolution.md Co-authored-by: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 8fe066b2..6827ca36 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -121,8 +121,6 @@ The default limiter is `VENKATAKRISHNAN`. ### Limiter Parameters and Further Details The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. -This is different than the small constant used to prevent division by zero, which is used in all these limiters. - For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). From 7e8abc12d59fc9a4e36d79a20a024d8e2a215628 Mon Sep 17 00:00:00 2001 From: Grant Norman <36522261+gnorman7@users.noreply.github.com> Date: Sun, 4 Dec 2022 13:50:41 -0700 Subject: [PATCH 17/22] Update _docs_v7/Slope-Limiters-and-Shock-Resolution.md Co-authored-by: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 6827ca36..31daf58b 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -125,7 +125,7 @@ The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting t For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). Thus, a large value will approach the case of using no limiter with undamped oscillations, while too small of a value will slow the convergence and add extra diffusion. -This value depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. +**Note:** This value depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. From a8a7cb283edad7ce415bb40d543be2fa80905d14 Mon Sep 17 00:00:00 2001 From: Grant Norman Date: Sun, 4 Dec 2022 14:24:49 -0700 Subject: [PATCH 18/22] PR, address @pcarruscag other comments on options --- .../Slope-Limiters-and-Shock-Resolution.md | 26 ++++++------------- 1 file changed, 8 insertions(+), 18 deletions(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 31daf58b..72a61fb1 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -106,15 +106,12 @@ The `SLOPE_LIMITER_` options above may each be changed to use different limiters | Type | Description | Notes | | --- | --- | --- | | `NONE` | No limiter | | -| `BARTH_JESPERSEN` | Barth-Jespersen | | +| `BARTH_JESPERSEN` | Barth-Jespersen | This limiter is a smooth version of the commonly seen Barth-Jespersen limiter seen in the literature | | `VENKATAKRISHNAN` | Venkatakrishnan | | | `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | | | `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification | This limiter should not be used for flow solvers | | `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | This limiter should not be used for flow solvers | -| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter has been observed to not output limiter values when called using VOLUME_OUTPUT | - - - +| `VAN_ALBADA_EDGE` | Van Albada (edge formulation) | This limiter is only implemented for flow solvers and does not output limiter values when using the VOLUME_OUTPUT option | The default limiter is `VENKATAKRISHNAN`. @@ -125,28 +122,21 @@ The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting t For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). Thus, a large value will approach the case of using no limiter with undamped oscillations, while too small of a value will slow the convergence and add extra diffusion. -**Note:** This value depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. - - - - - - - -Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`. -The `REF_ELEM_LENGTH` parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters. +The SU2 implementation of the `BARTH_JESPERSEN` limiter actually uses `VENKATAKRISHNAN` with $$K=0$$. +**Note:** the value of `VENKAT_LIMITER_COEFF` depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. Note that this global operation may incur extra time costs due to communication between MPI threads. The original work by [Wang](https://doi.org/10.2514/6.1996-2091) suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. +**Note:** unlike the aforementioned `VENKATAKRISHNAN` limiter, the `VENKATAKRISHNAN_WANG` limiter does not depend directly on the mesh size and can thus be used without non-dimensionalization. If the `VENKATAKRISHNAN` limiter is used outside of non-dimensional mode, the fields with larger values (pressure and temperature) will generally be limited more aggressively than velocity. + The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**. A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**. For both the `SHARP_EDGES` and `WALL_DISTANCE` limiters, the influence of the geometric modifier is controlled with `ADJ_SHARP_LIMITER_COEFF` which defaults to 3.0. -**Note:** these limiters should not be used for flow solvers. - - +**Note:** these limiters should not be used for flow solvers, as they only apply to the continuous adjoint solvers. + Increasing this parameter will decrease the value of the limiter and thus make the field more diffusive and less oscillatory near the feature (sharp edge or wall). In the `SHARP_EDGES` limiter, the qualification of what makes an edge "sharp" is described by the parameter `REF_SHARP_EDGES` (defaults to 3.0). Increasing this will make more edges qualify as "sharp". From e39d4de3dd47caed8389dd4a36b2b6a0f024cb5f Mon Sep 17 00:00:00 2001 From: kamo6346 <112155073+kamo6346@users.noreply.github.com> Date: Sun, 4 Dec 2022 21:23:11 -0700 Subject: [PATCH 19/22] Update _docs_v7/Slope-Limiters-and-Shock-Resolution.md Co-authored-by: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 72a61fb1..55405a08 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -28,7 +28,7 @@ For many studying compressible flow or high-speed aerodynamics, the formation of Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. -SU2 uses **slope limiters** to avoid these oscillations by switching to a low-resolution scheme near the shock, while switching back to a high-resolution scheme where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and ensures numerical stability in regions close to the shock. +SU2 uses **slope limiters** to avoid these oscillations by damping second-order terms near shocks and other regions with sharp gradients. The second-order reconstruction is kept where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and helps obtain physical results and numerical stability in regions close to the shock. Before mathematically describing the form of the limiters implemented in SU2, it is useful to briefly understand two concepts. These include **Total Variation** and **Godunov's Theorem**. From cf69af123a38c0f5fb497583c079b8727b063f0c Mon Sep 17 00:00:00 2001 From: kamo6346 <112155073+kamo6346@users.noreply.github.com> Date: Sun, 4 Dec 2022 21:24:39 -0700 Subject: [PATCH 20/22] Update _docs_v7/Slope-Limiters-and-Shock-Resolution.md Co-authored-by: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 55405a08..1efcc360 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -22,7 +22,7 @@ The options listed here do not apply to the high order DG solver. ## Theory: An introduction to slope limiters -For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities are a common occurrence. The use of high-order numerical schemes are desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flowfield. However, linear high-resolution schemes often result in numerical oscillations near the shock due to high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that greatly degrade the accuracy of your solution and pollute the domain. An example of this phenomena is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities is a common occurrence. The use of high-order numerical schemes is desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flow field. However, high-resolution linear schemes often result in numerical oscillations near the shock due to the high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that significantly degrade the accuracy of your solution and pollute the domain. An example of this phenomenon is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. From 41811bc2647f995581965085bf92cddf8d4e4aa7 Mon Sep 17 00:00:00 2001 From: kymonroe Date: Sun, 4 Dec 2022 21:28:15 -0700 Subject: [PATCH 21/22] Added references that example figures were created in 'MATLAB' in according to suggestions --- _docs_v7/Slope-Limiters-and-Shock-Resolution.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 1efcc360..6405bd05 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -22,11 +22,11 @@ The options listed here do not apply to the high order DG solver. ## Theory: An introduction to slope limiters -For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities is a common occurrence. The use of high-order numerical schemes is desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flow field. However, high-resolution linear schemes often result in numerical oscillations near the shock due to the high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that significantly degrade the accuracy of your solution and pollute the domain. An example of this phenomenon is shown below with the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. +For many studying compressible flow or high-speed aerodynamics, the formation of shock discontinuities is a common occurrence. The use of high-order numerical schemes is desired to resolve these regions as the strength of the shock largely governs the behavior of the downstream flow field. However, high-resolution linear schemes often result in numerical oscillations near the shock due to the high-frequency content associated with the shock. These oscillations can result in non-physical values (e.g. negative density) that significantly degrade the accuracy of your solution and pollute the domain. An example of this phenomenon is shown below with a MATLAB implementation of the Lax-Wendroff scheme for scalar advection. Although the Lax-Wendroff method is second-order, note that it introduces numerical oscillations that result in the state value of $$u$$ becoming negative. -Figure (1): A one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. +Figure (1): A MATLAB simulation of a one period advection (red) of an initial value discontinuity (black) using the Lax-Wendroff method. SU2 uses **slope limiters** to avoid these oscillations by damping second-order terms near shocks and other regions with sharp gradients. The second-order reconstruction is kept where the solution is smooth. This preserves solution accuracy in regions with smooth gradients and helps obtain physical results and numerical stability in regions close to the shock. @@ -176,7 +176,7 @@ SU2_MPI::Error("Unknown limiter type.", CURRENT_FUNCTION); --> ## Empirical comparison of limiters on a periodic advective domain -An example problem of the linear advection problem against four unique wave-forms was simulated to illustrate differences between the primary limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: +An example problem of the linear advection problem against four unique wave-forms was simulated in MATLAB to illustrate differences between the primary limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case: $$ u_j^{n+1} = u_j^{n} - \sigma (u_j^{n} - u_{j-1}^{n}) - \frac{1}{2}\sigma(1-\sigma) \left[ \phi_{j+\frac{1}{2}}(u_{j+1}^{n} - u_j^{n}) - \phi_{j-\frac{1}{2}}(u_{j}^{n} - u_{j-1}^{n}) \right] $$ @@ -184,7 +184,7 @@ where $$\phi_{j+\frac{1}{2}}$$ is the scalar value of the limiter at the interfa -Figure (3): A one period advection (red) of an initial condition (black) using various schemes, with and without limiters. +Figure (3): A MATLAB simulation of one period advection (red) of an initial condition (black) using various schemes, with and without limiters. From the above example we note: * The **Lax-Wendroff** scheme produces oscillations near sudden gradients due to dispersion errors. From Godunov's theorem this is expected as the scheme is second-order accurate and does not utilize a limiter. From 3933f0252c2349ba5db5868c257995197ebc022f Mon Sep 17 00:00:00 2001 From: Pedro Gomes <38071223+pcarruscag@users.noreply.github.com> Date: Mon, 5 Dec 2022 10:11:46 -0800 Subject: [PATCH 22/22] Apply suggestions from code review --- .../Slope-Limiters-and-Shock-Resolution.md | 34 ++----------------- 1 file changed, 3 insertions(+), 31 deletions(-) diff --git a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md index 6405bd05..019c40db 100644 --- a/_docs_v7/Slope-Limiters-and-Shock-Resolution.md +++ b/_docs_v7/Slope-Limiters-and-Shock-Resolution.md @@ -63,16 +63,11 @@ Recall that the original motivation for a slope limiter was to prevent the forma The inclusion of a slope limiter into a TVD scheme accomplishes this idea. - - - - - ## Available Limiter Options The field `SLOPE_LIMITER_FLOW` in the `.cfg` file specifies which limiter to use. Note that this option is only used if `MUSCL_FLOW = YES` (which specifies to use a second-order method). The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this. -The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations, rather than for the flow equations. +The [Turbulent Flat Plate example](https://su2code.github.io/tutorials/Turbulent_Flat_Plate/) sets `SLOPE_LIMITER_TURB`, which is used for the turbulence equations (if `MUSCL_TURB = YES`), rather than for the flow equations. More possible applications of limiters are listed below. @@ -87,14 +82,6 @@ More possible applications of limiters are listed below. | `SLOPE_LIMITER_ADJTURB` | Adjoint turbulence equations | Need `MUSCL_ADJTURB = YES` | - - The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below. **Note:** the Discontinuous-Galerkin methods (DG) / Higher-order methods (HOM) do not use limiters. @@ -119,14 +106,14 @@ The default limiter is `VENKATAKRISHNAN`. The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used. -For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size. +For the `VENKATAKRISHNAN`, `SHARP_EDGES`, and `WALL_DISTANCE` limiters, the `VENKAT_LIMITER_COEFF` parameter refers to $$K$$ in $$\epsilon^2=\left(K\bar{\Delta} \right)^3$$, where $$\bar{\Delta}$$ is an average grid size (this is hardcoded as 1m and thus all tuning is via $$K$$). The $$K$$ parameter defines a threshold, below which oscillations are not damped by the limiter, as described by [Venkatakrishnan](https://doi.org/10.1006/jcph.1995.1084). Thus, a large value will approach the case of using no limiter with undamped oscillations, while too small of a value will slow the convergence and add extra diffusion. The SU2 implementation of the `BARTH_JESPERSEN` limiter actually uses `VENKATAKRISHNAN` with $$K=0$$. **Note:** the value of `VENKAT_LIMITER_COEFF` depends on both the mesh and the flow variable and thus should be reduced if the mesh is refined. When using the `VENKATAKRISHNAN_WANG` limiter, `VENKAT_LIMITER_COEFF` is instead $$\varepsilon '$$ in $$\varepsilon = \varepsilon ' (q_{max} - q_{min})$$, where $$q_{min}$$ and $$q_{max}$$ are the respective *global* minimum and maximum of the field variable being limited. -Note that this global operation may incur extra time costs due to communication between MPI threads. +This global operation incurs extra time costs due to communication between MPI ranks. The original work by [Wang](https://doi.org/10.2514/6.1996-2091) suggests using `VENKAT_LIMITER_COEFF` in the range of $$[0.01, 0.20]$$, where again larger values approach the case of using no limiter. **Note:** unlike the aforementioned `VENKATAKRISHNAN` limiter, the `VENKATAKRISHNAN_WANG` limiter does not depend directly on the mesh size and can thus be used without non-dimensionalization. If the `VENKATAKRISHNAN` limiter is used outside of non-dimensional mode, the fields with larger values (pressure and temperature) will generally be limited more aggressively than velocity. @@ -160,21 +147,6 @@ After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen in the discrete adjoint formulation (default is `NO`). - - - ## Empirical comparison of limiters on a periodic advective domain An example problem of the linear advection problem against four unique wave-forms was simulated in MATLAB to illustrate differences between the primary limiters in SU2. The wave forms contain both smooth and discontinuous initial conditions and are advected for a single period with a CFL of $$\sigma = 0.8$$. The domain is discretized with $$N = 200$$ cells. The Lax-Wendroff scheme was used as a comparative case:

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