Compact multi-stage reconstruction method on polyhedral unstructured grids: Extension to higher-order finite volume scheme

Abstract

In this paper, a novel high-order finite volume scheme is proposed for arbitrary unstructured grids based on multi-stage reconstruction procedure using a compact stencil. Unlike the conventional approaches such as k-exact formulation, the coefficients of the reconstruction polynomial are determined by using a deferred correction algorithm which is built on the relations that make use of not only the volume integrated averages but also the spatial derivatives on the neighboring cells. The newly designated FVMS4 (Finite Volume method based on Multi-Stage procedure with 4th-order reconstruction) substantially improves the solution accuracy in contrast to its previous variant, FVMS3, without extending the reconstruction stencil, which allows FVMS4 to be formulated with algorithmic simplicity and numerical efficiency. In order to eliminate the numerical oscillations associated with high-order schemes, it is combined with a limiting projection scheme AOD (Adaptive Order Detection) to choose the optimal degree of reconstruction polynomial for trouble cells that may contain the discontinuous solutions. The resulting reconstruction method, namely FVMS4-AOD scheme achieves essentially non-oscillatory, less-dissipative and highly accurate results for smooth and discontinuous solutions. Its performance has been verified by solving linear and nonlinear hyperbolic systems for various benchmark tests in 2 and 3 dimensions. The numerical results substantiate that the proposed method provides an innovative numerical formulation capable of handling both complex physics and geometries which has significant benefits for practical applications.