OPTIMAL L-1-RATE OF CONVERGENCE FOR THE VISCOSITY METHOD AND MONOTONESCHEME TO PIECEWISE-CONSTANT SOLUTIONS WITH SHOCKS

We derive optimal error bounds for the viscosity method and monotone d ifference schemes to an initial-value problem of scalar conservation l aws with initial data being a finite number of piecewise constants, su bject to the initial discontinuities satisfying the entropy conditions . It is known that the entropy solution of the problem is piecewise co nstant with a finite number of interacting shocks satisfying the entro py conditions. A rigorous analysis shows that both the viscosity metho d and monotone schemes to approach the initial-value problem have unif orm L-1-error bounds of O(epsilon) and O(Delta x) for the time +infini ty > t > 0, respectively, where epsilon and Delta x are their correspo nding viscosity coefficient and discrete mesh length. The results are improvements over the half-order rates of L-1-convergence. Numerical e xperiments for the Lax-Friedrichs scheme are presented and numerical r esults justify the theoretical analysis.