Singular Approximations to Hyperbolic Systems of Conservation Laws

Abstract

Consider an n × n hyperbolic system of conservation laws of the form

ut + f(u)x = 0,  (t,x) ∈ ℝ+ × ℝ, u ∈ ℝn.

Here u = (u1,…,un) is the vector of conserved quantities, while the components of f = (f1,…,fn) are the luxes. The system is said strictly hyperbolic if at each point u the Jacobian matrix Df(u) has n real, distinct eigenvalues

λ1(u)< ··· < λn(u).

A fundamental ingredient to prove existence and stability in BV is the introduction of a functional, the Glimm–Liu interaction functional, which controls the interactions among non linear waves.
Aim of this note is to present a simple interpretation of (the scalar part of) the interaction functional, and show how it can be extended to the following equation:

1. a parabolic equation of the form

   ut + f(u)x = uxx;

2. scalar semidiscrete schemes, for example the upwind scheme

   ut(t,x) + (f(u(t,x)) − f(u(t,x −1)) = 0,

   or the backward scheme

   (u(t,x) − u(t −1,x)) + f(u(t,x)) = 0;

3. 2 × 2 relaxation approximation, in particular

   ut + vx = 0,

   vt + ux = f(u) − v.

All these approximations are interesting from the physical and numerical point of view. Finding an interaction is one of the key steps toward the proof of BV bounds.