Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
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DOI：
10.1007/978-3-030-60610-7_3
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发表时间：
2020-05
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影响因子：
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通讯作者：
A. R. Winters;D. Kopriva;G. Gassner;F. Hindenlang
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文献类型：
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作者：
A. R. Winters;D. Kopriva;G. Gassner;F. Hindenlang
间断Galerkin（DG）方法由于其高阶精度和几何灵活性，在计算物理和工程中有着悠久的历史，用于偏微分方程的近似求解。然而，DG并不完美，还存在一些问题。关于鲁棒性，DG在过去的七年里经历了广泛的转变，成为其现代形式，提供了关于线性和非线性问题的解有界性的陈述。本章采用建设性的方法介绍了一个现代化的DG谱元方法的可压缩Navier-Stokes方程在三维曲线的背景下。数值格式的基础来自于谱方法的经典原理，包括多项式逼近和高斯型求积。我们确定混叠的鲁棒性问题的一个根本原因，经典DG谱方法。消除所述混叠误差需要特定的微分矩阵和控制方程中对流通量项的仔细离散化。
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.