Two-Dimensional Modification of Godunov’s Method of the Fourth Order in Space and the Third Order in Time

A modification of Godunov’s method for two-dimensional nonstationary gas dynamics equations is presented. It has the fourth order of approximation in space and the third order in time. The difference scheme of the method is based on the joint discretization of the equations in space and time without using Runge–Kutta stages; i.e., it is completely discrete. The fluxes are calculated as a result of solving the Riemann problem with corrections to its arguments. New versions of TVD limiters of central differences are proposed, and they are applied to derivatives higher than the second order of accuracy. Results of an experimental verification of the approximation order of the method on two-dimensional smooth solutions within the Riemann and Prandtl–Meyer expansion fans are presented. A comparison with other methods is made both in terms of accuracy and performance.