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We derive a priori error of the Godunov method for the multidimensional Eulersystem of gas dynamics. To this end we apply the relative energy principle andestimate the distance between the numerical solution and the strong solution.This yields also the estimates of the $L^2$-norm of errors in density, momentumand entropy. Under the assumption that the numerical density and energy arebounded, we obtain a convergence rate of $1/2$ for the relative energy in the$L^1$-norm. Further, under the assumption -- the total variation of numericalsolution is bounded, we obtain the first order convergence rate for therelative energy in the $L^1$-norm. Consequently, numerical solutions (density,momentum and entropy) converge in the $L^2$-norm with the convergence rate of$1/2$. The numerical results presented for Riemann problems are consistent withour theoretical analysis.

Error estimates of the Godunov method for the multidimensional compressible Euler system