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Finished draft of describing SU2 options.
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‎_docs_v7/Gradients-Limiters-Grant.md‎

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* general documenation logistics: building, markdown / html specifics, github pull requests
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## Basics of describing what options are available and providing some references for them
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Do this before diving into the comparisons (which can be a lot more work)
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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.
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<!-- 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. -->
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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).
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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.
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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).
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The [Laminar Cylinder](https://su2code.github.io/tutorials/Laminar_Cylinder/) shows an example of this.
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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.
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More possible applications of limiters are listed below.
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<!-- There's also:
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% Frozen the slope limiter in the discrete adjoint formulation (NO, YES)
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FROZEN_LIMITER_DISC= NO
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-->
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<!-- Do I need this as a title? -->
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Slope Limiter Fields
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| Configuration Field | Description | Notes |
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| --- | --- | --- |
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| `SLOPE_LIMITER_FLOW` | Flow equations | Need `MUSCL_FLOW = YES` |
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| `SLOPE_LIMITER_TURB` | Turbulence equations | Need `MUSCL_TURB = YES` |
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| `SLOPE_LIMITER_SPECIES` | Species evolution equations | Need `MUSCL_SPECIES = YES` |
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| `SLOPE_LIMITER_ADJFLOW` | Adjoint flow equations | Need `MUSCL_ADJFLOW = YES` |
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| `SLOPE_LIMITER_ADJTURB` | Adjoint turbulence equations | Need `MUSCL_ADJTURB = YES` |
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<!--
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-->
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The `SLOPE_LIMITER_` options above may each be changed to use different limiters, as explained below.
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The `SLOPE_LIMITER_` options above may each be changed to use different limiters, which are listed and explained below.
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All of the limiters depend on variables `proj`, `delta`, and `eps`/`eps2` (and maybe `dist` for `raisedSine` in `SHARP_EDGES`)
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Available Limiters
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Limiters
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| Type | Description | Notes |
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| --- | --- | --- |
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| `NONE` | No limiter | |
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| `BARTH_JESPERSEN` | Barth-Jespersen | |
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| `VENKATAKRISHNAN` | Venkatakrishnan | |
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| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang | |
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| `SHARP_EDGES` | Venkatakrishnan with sharp edge modification | |
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| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification | |
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| Type | Description |
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| --- | --- |
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| `NONE` | No limiter |
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| `BARTH_JESPERSEN` | Barth-Jespersen |
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| `VENKATAKRISHNAN` | Venkatakrishnan |
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| `VENKATAKRISHNAN_WANG` | Venkatakrishnan-Wang |
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| `SHARP_EDGES` | Venkatakrishnan with sharp-edge modification |
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| `WALL_DISTANCE` | Venkatakrishnan with wall distance modification |
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| `VAN_ALBADA_EDGE` | [^1] Van Albada (edge formulation) |
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The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $0.05$, but its specific definition depends on the limiter being used.
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[^1]: This limiter may or may not be implemented for certain solvers. It may also suffer from problems of not outputing limiter values.
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<!-- TODO: Kal, maybe clarify / add some details to the above? -->
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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.
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The `VENKAT_LIMITER_COEFF` parameter is generally a small constant, defaulting to $$0.05$$, but its specific definition depends on the limiter being used.
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This is different than the small constant used to prevent division by zero, which is used for all limiters.
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[Wang](https://doi.org/10.2514/6.1996-2091)
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<!-- Maybe a better way to word this. -->
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After the number of iterations given by `LIMITER_ITER` (default $999999$), the value of the limiter will be frozen.
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<!-- We can specify which limiters are applied through the fields
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constexpr size_t MAXNVAR = 32; -->
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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.
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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).
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Thus, a large value will approach the case of using no limiter, while too small of a value will slow the convergence.
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This value depends on both the mesh and the flow variable.
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<!-- maybe change wording from flow variable to "field variable being limited" -->
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<!-- ??? should this section be included ??? -->
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<!-- so, \bar{\Delta} is actually config.GetRefElemLength(), which refers to RefElemLength -->
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Similarly, the parameter `REF_ELEM_LENGTH` controls $$\bar{\Delta}$$, but the behavior of the limiter should be controlled through `VENKAT_LIMITER_COEFF`.
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This parameter is also used in the geometric factor of the `SHARP_EDGES` and `WALL_DISTANCE` limiters.
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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.
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Note that this global operation may incur extra time costs due to communication between MPI threads.
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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.
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The `NONE`, `BARTH_JESPERSEN`, `VENKATAKRISHNAN`, and `VENKATAKRISHNAN_WANG` limiter options all have no **geometric modifier**.
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A geometric modifier increases limiting near walls or sharp edges. This is done by multiplying the limiter value by a **geometric factor**.
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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.
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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).
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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".
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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).
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<!-- ??? Are these limiters only for adjoints???!!! -->
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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`.
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$$ d(d_{\text{feature}}; c, K) = \frac{d_{\text{feature}}} { (c \cdot K \bar{\Delta}) } - 1$$
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Then, the geometric factor is given by
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$$ \gamma (d) = \frac{1}{2} (1+d+\sin(\pi \cdot d)/ \pi) $$
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Note that the geometric factor is nonnegative and nondecreasing in $d_{feature}$.
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<!-- Maybe missing some subsection / transitions here. -->
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<!-- Maybe a better way to word this. -->
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After the number of iterations given by `LIMITER_ITER` (default $$999999$$), the value of the limiter will be frozen.
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The option `FROZEN_LIMITER_DISC` tells whether the slope limiter is to be frozen in the discrete adjoint formulation (default is `NO`).
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<!-- We can specify which limiters are applied through the fields
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constexpr size_t MAXNVAR = 32; -->
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Should I mention some possible errors
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-->
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## Why Slope Limiters are used in FVM
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* TVD
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* Monotonic
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## Mathematically describe limiters available to user in SU2
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Also discuss their properties.
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<!-- We need to first figure out what is going on with:
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1) venkatFunction, missing '2'
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2) Barth-Jespersen using venkatFunction and not non-smooth min/max
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3) Van Albada... just... what?
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-->
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<!-- TODO: Kal and Grant need to resolve notation, ex: k vs. K -->
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## Empirical comparison of the available limiters on a test problem
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Flowfield colored by the limiter value.
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Link to the Documentation on how to generate these

‎_docs_v7/Gradients-Limiters-Kal.md‎

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<!-- high order == high accuracy, maybe change wording KM: changed wording -->
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<!-- oscillations can result **in** non-physical values KM: fixed grammer-->
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<!-- second order to second-order ? KM: fixed to add hyphen -->
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<!-- ??? "second order" may actually be correct when "order" is used as a noun.
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Definitely "second-order" when used as an adjective, but not sure if "second-order" when used as a noun. -->
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<img src="../../docs_files/LW_example.png" width="500">
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