Numerical solution of test problems using a modified Godunov scheme

Modifications of Godunov’s scheme based on V.P. Kolgan’s approach to the construction of second-order accurate schemes in space for smooth solutions are proposed. The gasdynamic parameters linearly interpolated in a mesh cell, but the Riemann problem is solved for parameters at an intermediate point between the center and the boundary of the cell. Properties of Kolgan’s scheme and the proposed modifications, such as monotonicity and entropy nondecrease, are examined as applied to the system of differential equations describing plane sound waves propagation in a resting gas. Test problems in nonlinear gas dynamics are solved, namely, the Riemann problem in a pipe, transformation of an inhomogeneity in a plane-parallel flow, supersonic flow entering an axisymmetric convergent-divergent nozzle with a coaxial central body, and coaxial supersonic flow around a cylinder. The efficiency of schemes with an intermediate point is demonstrated.