4.1 High-resolution shock-capturing schemes

High-resolution methods: modified high order methods with the appropriate amount of numerical dissipation in the vicinity of a discontinuity

Let us consider the following (Cauchy) IVP

$$\frac{\partial u}{\partial t} + \frac{\partial f(u)} {\partial x} = 0 \hspace{1cm} u(x,0)=u_0(x)$$ (97)

A finite difference scheme is a time-marching procedure which permits to obtain approximations to the solution in the mesh points, $u_j^{n+1}$, from the approximations in previous time steps $u_j^{n}$

Quantity $u_j^{n}$ is an approximation to $u(x_j,t^n)$ but, in the case of a conservation law, it is often preferable to view it as an approximation to the average of $u(x,t)$ within the numerical cell $[x_{j-1/2},x_{j+1/2}]$ ( $x_{j \pm 1/2} = (x_j + x_{j \pm 1})/2$)

$$u_j^n \approx \frac{1}{\Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}} u(x,t^n) dx,$$ (98)

consistent with the integral form of the conservation law

For hyperbolic systems of conservation laws, schemes written in conservation form guarantee that the convergence (if it exists) is to one of the weak solutions of the original system of equations (Lax-Wendroff theorem 1960).

An algorithm written in conservation form reads

$$u_j^{n+1} = u_j^n - \frac{\Delta t}{\Delta x} \left( \hat{f}(u_{j-r}^n,u_{j-r+1}^n,\ldots,u_{j+q}^n) \right.$$

$$\left. \hspace{1.8cm} - \hat{f}(u_{j-r-1}^n,u_{j-r}^n, \ldots,u_{j+q-1}^n) \right)$$ (99)

where $\hat{f}$ is a consistent numerical flux function

$$\hat{f}(u,u,\ldots,u) = f(u)$$ (100)

The Lax-Wendroff theorem does not state whether the method converges.

To guarantee convergence, some form of stability is required, as for linear problems (Lax equivalence theorem, see Ritchmyer & Morton 1967)

The notion of total-variation stability has proven very successful although powerful results have only been obtained for scalar conservation laws

The total variation of a solution at $t=t^n$, TV($u^n$), is defined as

$${\rm TV}(u^n) = \sum_{j=0}^{+\infty} \vert u_{j+1}^n - u_j^n\vert$$ (101)

and a numerical scheme is said to be TV-stable if TV($u^n$) is bounded for all $\Delta t$ at any time for each initial data

A convergence theorem for non-linear, scalar, conservation laws (LeVeque 1991): For numerical schemes in conservation form with consistent numerical flux functions, TV-stability is a sufficient condition for convergence

A current line of research focuses on developing high-order, accurate methods in conservation form satisfying the condition of TV-stability

The conservation form is ensured by starting with the integral version of the partial differential equation in conservation form. Integrating the PDE within a space-time computational cell $[x_{j-1/2},x_{j+1/2}]\times[t^n,t^{n+1}]$, the numerical flux function $\hat{f}_{j+1/2}$ is seen to be an approximation to the time-averaged flux across the interface, i.e.,

$$\hat{f}_{j+1/2} \approx \frac{1}{\Delta t} \int_{t^n}^{t^{n+1}} f(u(x_{j+1/2},t)) dt$$ (102)

The flux integral depends on the solution at the numerical interfaces, $u(x_{j+1/2},t)$, during the time step

Key idea: a possible procedure is to calculate $u(x_{j+1/2},t)$ by solving Riemann problems at every cell interface to obtain

$$u(x_{j+1/2},t) = u(0;u_j^n,u_{j+1}^n)$$ (103)

$u(0;u_j^n,u_{j+1}^n)$ denotes the Riemann solution for the (left and right) states $u_j^n$, $u_{j+1}^n$ along the ray $x/t=0$