Hyperbolic Relaxation Approximation to Nonlinear

Hyperbolic Relaxation Approximation to Nonlinear Parabolic Problems Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani

Abstract. A general idea for solving hyperbolic systems of conservation laws is to use a local relaxation approximation. The motivation is to have a simple discrete velocity kinetic type relaxation regularization which approximates the original system with a small dissipative correction. In this paper we extend the previous approach to systems of nonlinear parabolic equations. The corresponding relaxation schemes are also constructed. The new approximation, while mantaining the advantages of that constructed for systems of conservation laws, at the cost of one more rate equation permits to transform second order nonlinear systems to semi-linear rst order ones.

1. Introduction

In this paper we study the numerical passage from an hyperbolic relaxation system towards the corresponding parabolic equilibrium limit equation. We also present a relaxation approximation to general systems of nonlinear convection-reactiondiusion equations [19, 20] which allows us to develop a class of stable numerical schemes for the original systems. In particular we show that for many of such systems the relaxing hyperbolic systems have a clear kinetic interpretation as generalized Carleman models [4] or Broadwell models [2] with source terms. The simplest prototype is given by the one-dimensional equation @t u + @x f (u) = @xx p(u) + s(u); (1) where (x; t) 2 R  R+ , p, f and s are given smooth functions such that p(0)  0 and p0 (u) > 0, with initial data u(x; 0) = u0 (x): (2) By introducing a new variable v, one can couple v and u through the following linear hyperbolic system @t u + @xv = s(u); (3) 1 1 @t v + 2 @x u = ? 2 k(u)(v ? f (u));

2

G. Naldi, L. Pareschi, and G. Toscani

with the additional initial condition v(x; 0) = v0 (x). Here  is a small positive parameter called the relaxation time and k(u) = (p0 (u))?1 . As usual when