Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System

Abstract

We derive a priori error estimates of the Godunov method for the multidimensional compressible Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the L^2-norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the L^1-norm, that is to say, a convergence rate of 1/4 for the L^2-error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the L^1-norm, consequently, the numerical solution converges in the L^2-norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis.