A NUMERICAL STUDY OF A ROTATIONALLY DEGENERATE HYPERBOLIC SYSTEM .2. THE CAUCHY-PROBLEMS

In this paper a numerical method is proposed for obtaining stable solu tions to the initial-value problem for the pair of conservation laws [ GRAPHICS] where U is an element of R(2). These equations constitute a standard model describing small amplitude plane wave solutions to rota tionally invariant partial differential equation (PDE) systems arising in continuum mechanics. In particular, viscous regularization of (0.1 ) by epsilon U-xx is not uniform as epsilon --> 0. Therefore finite di fference schemes, which include numerical dissipation of this type, fa il to approximate certain dynamically stable weak solutions of the inv iscid system (0.1). Our proposal is to extend the system and solve the three conservation laws [GRAPHICS] with data (r(0), U-0) satisfying r (0) = \U-0\. The solutions (r(epsilon), U-epsilon) of the viscous regu larization of (0.2) by a right-hand side (epsilon r(xx), epsilon U-xx) are shown to converge as the viscosity coefficient epsilon tends to 0 . It is also shown that the solutions (r(l), U-l) of the Lax-Friedrich s and Godunov finite difference schemes for (0.2) converge as the grid spacing l approaches 0. In all three cases, the unique limit (r, U) s olves (0.2), satisfies \U\ = r, and thus provides the dynamically stab le solution U of (0.1).