A new high-order spectral difference method for simulating viscous flows on unstructured grids with mixed-element meshes

• Published in: Computers & fluids Vol. 184; no. C; pp. 187 - 198
• Main Authors: Li, Mao, Qiu, Zihua, Liang, Chunlei, Sprague, Michael, Xu, Min, Garris, Charles A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 30.04.2019; Elsevier BV; Elsevier
• Subjects: Continuity (mathematics); Fluxes; MATHEMATICS AND COMPUTING; Mixed elements; Navier-Stokes equations; Quadrilaterals; Raviart-Thomas space; Spectra; Spectral difference method; Stability analysis; Unstructured grids (mathematics); Spectral difference method; Raviart-Thomas space; Mixed elements; Navier-Stokes equations
• ISSN: 0045-7930; 1879-0747; 1879-0747
• DOI: 10.1016/j.compfluid.2019.03.010

•Analyzed and proved linear stability of SRT schemes for mixed elements.•Pointed out a weak instability of SDRT scheme higher than 4th order.•Demonstrated stability and accuracy of SDRT method on meshes with mixed elements.•Extended SDRT method application to simulate both inviscid and viscous flows. 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 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. [1] and implemented on triangular-element meshes for invisid flow only. Our present results demonstrated that the SDRT method is stable and high-order accurate in two dimensions (2D) for a number of test problems by using triangular-, quadrilateral-, and mixed-element meshes for both inviscid and viscous flows. A stability analysis is also performed for mixed elements. We found our current SDRT scheme is stable for fourth-order accurate spatial discretizations. However, a stable fifth-order SDRT scheme remains to be identified.