Construction of modern robust nodal discontinuous Galerkin spectral element methods for the compressible Navier-Stokes equations. (English) Zbl 1472.76066
Kronbichler, Martin (ed.) et al., Efficient high-order discretizations for computational fluid dynamics. Selected papers based on the presentations at the summer school, Udine, Italy, July 16–20, 2018. Cham: Springer. CISM Courses Lect. 602, 117-196 (2021).
Summary: Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems. This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier-Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.
For the entire collection see [Zbl 1468.76003].
For the entire collection see [Zbl 1468.76003].
MSC:
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 76N06 | Compressible Navier-Stokes equations |
Keywords:
Legendre polynomial approximation; Gaussian quadrature; differentiation matrix; advective fluxReferences:
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