Symmetric difference schemes of componentwise splitting and equivalent predictor-corrector scheme based on the Godunov method as applied to multidimensional gasdynamic simulation

An approach is described for improving the accuracy of numerical solutions to multidimensional gasdynamic problems produced by Godunov's schemes. The basic idea behind the approach is to construct symmetric difference schemes based on splitting with respect to spatial variables with the subsequent transformation into equivalent predictor-corrector schemes. It is shown that the computation of “large” values by solving the one-dimensional Riemann problem at the interface of two neighboring cells leads to approximation errors in Godunov’s schemes. It is proposed to reconstruct large values so as to eliminate this source of errors. The time integration step in the modified schemes is consistent with that in the one-dimensional schemes and, on spatially uniform meshes, is 2 and 3 times larger than that in Godunov's classical schemes for two- and three-dimensional problems, respectively. The numerical results obtained for test problems confirm the improvement of the accuracy of solutions produced by the modified schemes.