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.
翻译:我们先验地得出了气体动态多维 Euler 系统Godunov 方法的误差。为此目的,我们应用相对能源原则,并估计数字溶液与强力溶液之间的距离。这也得出密度、动力和英特罗比误差以0.2美元为单位的估算值。根据数字密度和能量受约束的假设,我们获得了1/2美元相对能量的汇合率。此外,根据这一假设,数字溶液的总变异是受约束的,我们获得了以$1美元为单位的相对能量第一级汇合率。因此,以1/2美元为单位的数字溶剂(密度、动力和酶)与1/2美元汇合率的数值溶液。里曼问题的数字结果与我们的理论分析是一致的。