Discrete Shock Profiles and Their Stability

Abstract

We discuss the existence and the asymptotic stability of shock profiles associated to finite difference schemes. Assuming that the shock is stationary, we look for steady solutions of the scheme. There are strong resemblances with traveling waves obtained by means of a parabolic approximation (the so-called viscous profiles). However, main differences occur when the shock is under-compressive, these being the counterpart of the differences between the dynamics of vector fields and the one of diffeomorphisms.

The stability is investigated by means of an Evans function, in the spirit of the recent analysis of viscous profiles by R. Gardner & K. Zumbrun [2]. However, the results differ qualitatively.

Let

∂ _t u + ∂ _x f (u) = 0, x ∈ R, t> 0, (1)

be a system of conservation laws, where f: u → R ⁿ is a vector field, u being a convex open set in Rⁿ. We shall assume the strict hyperbolicity in the vicinity of distinguished points, u ^l and u^r, but this property may fail elsewhere, with the consequence that under-compressive (as well as over-ones) shocks may occur.

We investigate how faithfull is a finite difference scheme in presence of a discontinuous solution of (1). It is natural to restrict to conservative schemes and, for the sake of simplicity, we shall consider only the simplest ones, where the numerical flux F _j-1/2 is given as a function of two points F(u_j-_1;u_j):

$$ u_j^{m + 1} = u_j^m + \sigma \left( {F_{j - 1/2}^m - F_{j + 1/2}^m} \right),{\text{ }}\sigma : = \frac{{\Delta t}}{{\Delta x}}.$$

(2)

One also assumes the consistency (F(a, a)≡ f (a)) and the stability of constant states, which relies upon a CFL condition. From now on, the ratio v will be kept fixed once for all.

We often regard the scheme as a mapping Ф from l ^∞ (Z) into itself:

$$ {U^{M + 1}} = :\Phi \left( {{U^m}} \right),{\text{ }}{U^m}: = \left( {u_j^m} \right)j \in z.$$

Then the CFL condition means that constant states are stable under l²-perturbations; they even are often nonlinearly stable with respect to suitable perturbations.