High-order accurate monotone compact running scheme for multidimensional hyperbolic equations

Monotone absolutely stable conservative difference schemes intended for solving quasilinear multidimensional hyperbolic equations are described. For sufficiently smooth solutions, the schemes are fourth-order accurate in each spatial direction and can be used in a wide range of local Courant numbers. The order of accuracy in time varies from the third for the smooth parts of the solution to the first near discontinuities. This is achieved by choosing special weighting coefficients that depend locally on the solution. The presented schemes are numerically efficient thanks to the simple two-diagonal (or block two-diagonal) structure of the matrix to be inverted. First the schemes are applied to system of nonlinear multidimensional conservation laws. The choice of optimal weighting coefficients for the schemes of variable order of accuracy in time and flux splitting is discussed in detail. The capabilities of the schemes are demonstrated by computing well-known two-dimensional Riemann problems for gasdynamic equations with a complex shock wave structure.