Stability of discrete shocks for difference approximations to systems of conservation laws

The asymptotic stability of weak discrete stationary shocks for systems of conservation laws in one space dimension is proved. The difference approximation should be conservative, dissipative, and kth order accurate in space with odd k. The problem is considered in a finite interval |x| less than or equal to l with appropriate boundary conditions, where l is large compared with the width of the shock layer epsilon(-1) = |u(R) - u(L)|(-1/k). The proof is based on the assumption that the corresponding continuous shocks for the scalar problem u(t) + uu(x) = -(ipartial derivative(x))(k+1)u are stable. The latter is known to be true for k = 1 and k = 3.