Hybrid Spectral Difference/Embedded MPWENO Method for Conservation Laws

Abstract

Recently, interest has been increasing towards applying high-order methods to engineering applications with complex geometries. As a result, a family of discontinuous high-order methods, such as Discontinuous Galerkin (DG), Spectral Volume (SV) and Spectral Difference (SD) methods, are under active development. These methods provide spectral-like results and are highly parallelizable due to local solution reconstruction within each cell. But, these methods suffer from Gibbs phenomenon near discontinuities. Artificial viscosity and sub-cell shock capturing method have been developed circumventing this problem. As an attempt towards applying a discontinuous high-order method for large scale engineering applications involving discontinuities in flows with complex geometries, a hybrid SD/embedded FV method is introduced by Choi. In this hybrid approach, structured finite volume cells are embedded in hexahedral elements containing discontinuity and highorder shock capturing scheme is used to overcome Gibbs phenomenon. In smooth flow regions away from discontinuities, the spectral difference method is employed. In this paper, the hybrid SD/embedded FV method is further investigated with a suite of test cases. In addition, the idea of embedding structured FV elements employed in the hybrid SD/embedded FV method is further extended to unstructured hexahedral grid and is introduced as the embedded structured element (ESE) framework for high-order method using unstructured hexahedral grid. The embedded structured element framework is workin-progress, but it shows promising results for applying high-order method for complex geometries. The error analysis and a suite of 1D and 2D test cases are presented further investigating the hybrid SD/embedded FV method using structured grid. One example employing the ESE framework is also included and discussed.