High order sub-cell finite volume schemes for solving hyperbolic conservation laws II: Extension to two-dimensional systems on unstructured grids

Abstract

In this paper, the second in a series, the high order sub-cell finite volume method is extended to two-dimensional hyperbolic systems on unstructured quadrilateral grids. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume method is applied to each of the sub-cells. The average values on the sub-cells belonging to current and face neighboring main cells are used to reconstruct a common polynomial distribution of a dependent variable on current main cell. This method can achieve high order accuracy using a compact stencil. The focus of this paper is to study the performance of the sub-cell finite volume method on two-dimensional unstructured quadrilateral grids and to verify that high order accuracy can be achieved using face neighboring sub-cells only. Fourier analysis is performed to analyze the dispersion and dissipation properties of the two-dimensional sub-cell finite volume schemes. To capture the discontinuities, the paper proposes a cell-based multi-dimensional limiting procedure using only face-neighboring main cells. Several benchmark test cases are simulated to validate the proposed sub-cell finite volume schemes and the multi-dimensional limiting procedure.