NONLINEAR STABILITY OF DISCRETE SHOCKS FOR SYSTEMS OF CONSERVATION-LAWS

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general mXm sys tems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the L(P)-norm for all p greater than or equal to 1, provided that the sum s of the initial perturbations equal zero. These results should shed l ight on the convergence of the numerical solution constructed by the L ax-Friedrichs scheme for the single-shock solution of system of hyperb olic conservation laws. If the Riemann solution corresponding to the g iven far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discret e shocks are nonlinearly stable in L(P) (P greater than or equal to 2) . These results are proved by using both a weighted estimate and a cha racteristic energy method based on the internal structures of the disc rete shocks and the essential monotonicity of the Lax-Friedrichs schem e.