An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems

Abstract

We present a high-order formulation for solving hyperbolic conservation laws using the Discon-tinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge-Kutta time discretization. Some results of higher-order adaptive refinement calculations are presented for in-viscid Rayleigh Taylor flow instability and shock reflexion problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities. 1. Introduction. The Discontinuous Galerkin Method (DGM) was initially introduced by Reed and Hill in 1973 [16] as a technique to solve neutron transport problems. Lesaint [13] presented the first numerical analysis of the method for a linear advection equation. However, the technique lay dormant for several years and has only recently become popular as a method for solving fluid dynamics or electromagnetic problems [4]. The DGM is somewhere between a finite element and a finite volume method and has many good features of both. Finite element methods (FEMs), for example, involve a double discretization. First, the physical domain

DOI: 10.1137/S00361445023830

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