  We derive a priori error of the Godunov method for the multidimensional 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 $L^2$-norm of errors in density, momentum
and entropy. Under the assumption that the numerical density and energy are
bounded, 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 numerical
solution is bounded, we obtain the first order convergence rate for the
relative 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 with
our theoretical analysis.
