4.1.3 High-resolution shock-capturing schemes

This is the approach followed by an important subset of shock-capturing methods, the so-called Godunov-type methods (Harten & Lax 1983, Einfeldt 1988)

These methods are written in conservation form and use Riemann solvers to compute approximations to $u(0;u_j^n,u_{j+1}^n)$

High-order of accuracy is achieved in two different ways:

• Flux-limiter methods
• Slope-limiter methods

A remark: artificial viscosity methods

• Idea: take a high order finite difference method (e.g., Lax-Wendroff) and add an ``artificial viscosity" term $Q$ to the hyperbolic equation

• damp spurious oscillations

• $Q$ must vanish as $\Delta t$, $\Delta x \rightarrow 0$

• introduced by von Neumann and Richtmyer (1950) in the context of the classical hydrodynamic equations. $Q$ is a term accompanying the pressure of the form

  
  

  $$Q = \left\{ \begin{array}{ll} -\alpha\frac{\partial v}{\par... ...\partial t}>0$} \\ 0 & \mbox{otherwise} \end{array} \right.$$ (106)

  
  

  with $\alpha=\rho(k\Delta x)^2\frac{\partial v}{\partial x}$

• Advantages:
  • easy to implement
  • efficient

• Disadvantages:
  • problem dependent
  • problematic for ultrarelativistic flows

Flux limiter methods:

The numerical flux is obtained from a high order flux (e.g., the Lax-Wendroff flux) in the smooth regions and from a low order flux (e.g., the flux from some monotone method) near discontinuities

$$\hat{f} = \hat{f}_h - (1-\Phi)(\hat{f}_h-\hat{f}_l)\nonumber$$

with the limiter $\Phi\in[0,1]$

Example: the flux-corrected-transport algorithm (Boris and Book 1973), one of the earliest high resolution methods

Slope limiter methods:

Use of conservative polynomial functions to interpolate the approximate solutions within the numerical cells

The idea is to produce more accurate left and right states for the Riemann problems by substituting the mean values $u_j^n$ (that give only first-order accuracy) for better approximations of the true flux near the interfaces, $u_{j+1/2}^{\rm L}$, $u_{j+1/2}^{\rm R}$

The interpolation algorithms have to preserve the TV-stability of the scheme and this is usually achieved by using monotonic functions which lead to a decrease in the total variation (total-variation-diminishing schemes, TVD, Harten 1984)

If R is an interpolant function for the approximate solution $u^n$ and $\tilde{u}(x,t^n)$ is the interpolated function within the cells, i.e., $\tilde{u}(x,t^n) = {\rm R}(u^n;x)$, satisfying ${\rm TV}(\tilde{u}(\cdot,t^n)) \leq {\rm TV}(u^n)$ then it can be proven that the whole scheme verifies ${\rm TV}(u^{n+1}) \leq {\rm TV}(u^n)$.

High-order TVD schemes were first constructed by van Leer (1979) who obtained second-order accuracy by using monotonic piecewise linear slopes for cell reconstruction

$$\begin{eqnarray*} {\bf u}^{L,R}_j = {\bf u}_j + s_j(x_{j\pm\frac{1}{2}}-x_j) \hspace{2cm} x_j=\frac{1}{2}(x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}) \end{eqnarray*}$$

$$\begin{eqnarray*} s_j= \left\{ \begin{array}{cc} \min(\Delta Q^{j+\frac{1}{2}},... ...^{j-\frac{1}{2}})$} \\ 0 & \mbox{otherwise} \end{array}\right. \end{eqnarray*}$$

$$\begin{eqnarray*} \Delta Q^{j+\frac{1}{2}}=\frac{{\bf u}_{j+1}-{\bf u}_j}{x_{j+1}-x_{j}} \end{eqnarray*}$$

The piecewise parabolic method (PPM) of Colella and Woodward (1984) provides third-order accuracy

The TVD property implies TV-stability but can be too restrictive. In fact, TVD methods degenerate to first order accuracy at extreme points (Osher & Chakravarthy 1984)

Other reconstruction alternatives have been developed in which some growth of the total variation is allowed:

• total-variation-bounded (TVB) schemes (Shu 1987)
• essentially nonoscillatory (ENO) schemes (Harten, Engquist, Osher & Chakravarthy 1987)
• piecewise-hyperbolic method (PHM) (Marquina 1994)