A new high-order spectral difference method for simulating viscous flows on unstructured grids with mixed-element meshes Open Access
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In the present study, a spectral difference (SD) method is developed for viscous flows on meshes with a mixture of triangular and quadrilateral elements. The SD method was originally developed by Liu et al. for simplex elements and has been studied for years. The standard SD method for triangular elements, which employs Lagrangian interpolating functions for fluxes, is not stable when the designed accuracy of spatial discretization is third order or higher. Unlike the standard SD method, the method examined here uses vector interpolating functions in the Raviart-Thomas (RT) spaces to construct continuous flux functions on reference elements. The spectral-difference Raviart-Thomas (SDRT) method was originally proposed by Balan et al. and implemented on triangular-element meshes for invisid flow only. In the presented study, a rigorous linear stability analysis of spatial and full discretization was performed on two dimensional (2D) quadrilateral RT elements and mixed elements. We found that the current SDRT scheme is stable for fourth order accurate spatial discretizations while a stable fifth order SDRT scheme remains to be identified. A new 2D SDRT solver was developed for compressible Navier-Stokes equations that further extended the application of SDRT method to quadrilateral- and mixed-element meshes for both inviscid and viscous flows. Through different test cases and comparative numerical studies, our obtained results verified the stability and high-order accuracy of SDRT method by using triangular-, quadrilateral-, and mixed-element meshes for both inviscid and viscous flows.