High order subcell finite volume method in solving hyperbolic conservation laws

Abstract

In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed. The basic idea of this method is to subdivide a control volume into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The cell averages on the sub-cells of current and face neighboring control volume are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume and PNPM methods but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume and PNPM methods and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. A modified multi-dimensional limiting procedure is constructed to capture the discontinuities. And several two-dimensional benchmark test cases are simulated to study the performances of the proposed schemes.