Optimal monotonization of a high-order accurate bicompact scheme for the nonstationary multidimensional transport equation

Abstract

A hybrid scheme is proposed for solving the nonstationary inhomogeneous transport equation. The hybridization procedure is based on two baseline schemes: (1) a bicompact one that is fourth-order accurate in all space variables and third-order accurate in time and (2) a monotone first-order accurate scheme from the family of short characteristic methods with interpolation over illuminated faces. It is shown that the first-order accurate scheme has minimal dissipation, so it is called optimal. The solution of the hybrid scheme depends locally on the solutions of the baseline schemes at each node of the space-time grid. A monotonization procedure is constructed continuously and uniformly in all mesh cells so as to keep fourth-order accuracy in space and third-order accuracy in time in domains where the solution is smooth, while maintaining a high level of accuracy in domains of discontinuous solution. Due to its logical simplicity and uniformity, the algorithm is well suited for supercomputer simulation.