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

摘要

We derive optimal error bounds for the viscosity method and monotone difference schemes to an initial-value problem of scalar conservation laws with initial data being a finite number of piecewise constants, subject to the initial discontinuities satisfying the entropy conditions. It is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions. A rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial-value problem have uniform L-1-error bounds of O(epsilon) and O(Delta x) for the time +infinity > t > 0, respectively, where epsilon and Delta x are their corresponding viscosity coefficient and discrete mesh length. The results are improvements over the half-order rates of L-1-convergence. Numerical experiments for the Lax-Friedrichs scheme are presented and numerical results justify the theoretical analysis. [References: 20]