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 high-order 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 work-in-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.
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