A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

Abstract

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the α-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the α-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.

A Numerical Study of Integrated Linear Reconstruction for Steady Euler Equations Based on Finite Volume Scheme

“…Steady Euler equations play an important role in aerodynamic shape optimization problems [1,14], and have been viewed as a benchmark problem to evaluate the performance of various numerical schemes in computational fluid dynamics (CFD), see [4,24,37,44,45] among others. A lot of numerical methods have been developed to solve the unsteady and steady Euler equations, e.g., the discontinuous Galerkin method [4,33,50,51], the finite volume method [3,22,36,44], the spectral volume method [47], and the fast sweeping method [11,12].…”