We present our recent result on the error analysis of the finite volume Godunov method when applied to multidimensional Euler equations of gas dynamics. The main tool is to use a problem-related metric, the so-called relative energy. This allows us to study a priori errors between a finite volume solution and a strong solution of the Euler system. Theoretical results will be illustrated by a series of numerical experiments.This is a join work with Bangwei She (Beijing) and Yuhuan Yuan (Nanjing). This research has been supported by the German Science Foundation under the TRR 146 grant.