Flux limiter - AST-Course/AST5110 GitHub Wiki

This documentation is a breif summary of Computational Gasdynamics book (sections 20-23), by Laney.

TVD schemes are immense, and here we will give only a concise sample to give an idea of this approach. TVD can be logically separated by flux-averaged (or flux-reconstructed) methods and solution-averaged methods (reconstruction-evolution methods). At the same time, the former can be divided into three groups: flux-limited, flux-corrected, and self-adjusting hybrid methods, but essentially all three are the same.

For linear systems of equations, flux averaging is the same as solution averaging. In other words, if $f(u) = a, u$ where $a$ is a constant, averaging to find $u$ and then finding $f(u)$ is the same as averaging to find $f(u)$ directly. However, this is usually not the same for non-linear equations with non-linear fluxes.

Flux-limited methods.

Flux-limited methods combine two so-called first-generation methods. Therefore, for a scalar conservative law, flux-limited methods are defined as follows:

$$u^{n+1} = u_i^{n} - \lambda (\hat{f}^n_{i+1/2}+\hat{f}^n_{i-1/2})$$

where

$$\hat{f}^{n}{i+1/2} = \hat{f}^{(1)}{i+1/2} + \phi^{n}{i+1/2}(\hat{f}^{(2)}{i+1/2}-\hat{f}^{(1)}_{i+1/2})$$

and $f^{(1)}{i+1/2}$ and $f^{(2)}{i+1/2}$ are the conservative numerical fluxes of two different first-generation methods with complementary properties, e.g., Roe's first-order method and Lax-Wendroff method. $\phi^n_{i+1/2}$ is the adaptive parameter controlling the linear combination of the flux limiter. Normally, the adaptive parameter jumps the spatial index up or down by $1/2$ depending on the wind direction. For instance, if $a>0$:

$$\hat{f}^{n}{i+1/2} = \hat{f}^{(1)}{i+1/2} + \phi^n_{i}(\hat{f}^{(2)}{i+1/2}-\hat{f}^{(1)}{i+1/2})$$

and if $a<0$:

$$\hat{f}^n_{i+1/2} = \hat{f}^{(1)}{i+1/2} + \phi^n{i+1/2}(\hat{f}^{(2)}{i+1/2}-\hat{f}^{(1)}{i+1/2})$$

$\phi$ is modulated depending on the two first-generation methods. One common approach is to have one method near shocks and a different one at smooth transitions. Flux-limited methods use as a reference of the two regions a difference of two neighboring grid points.

Flux-corrected methods.

While Flux-limited methods average two methods, Flux-corrected methods alter one method to me it more like another method:

$$\hat{f}^{n}{i+1/2} = \hat{f}^{(1)}{i+1/2} + \hat{f}^{(C)}{i+1/2} = \hat{f}^{(1)}{i+1/2} + diff^n_{i}(\hat{f}^{(1)}{i+1/2},\hat{f}^{(2)}{i+1/2})$$

$\hat{f}^{(C)}_{i+1/2}$ is the corrective numerical flux.

Both Flux methods are very similar to that end. In (exercise 4)[https://github.com/AST-Course/AST5110/blob/main/ex_4a.ipynb], who will look like $\hat{f}^{(C)}_{i+1/2}$ to be a flux-limited?

Self-adjusting hybrid methods.

It combines two complementary methods and is as follows:

$$\hat{f}^{n}{i+1/2} = \theta^n{i+1/2}\hat{f}^{(1)}{i+1/2} + (1-\theta^n{i+1/2})\hat{f}^{(2)}_{i+1/2}$$

where $\theta^n_{i+1/2}$ is known as schock switch. This term should be near 1 at shocks and 0 in smooth regions so that this term will distinguish the two regions. Just to let you know, it is impossible to design a perfect shock switch.

Reconstruction-evolution methods.

Not covered in the course.