Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology

The compact form of the discontinuous Galerkin method allows for a detailed local analysis of the method in the neighborhood of the shock for a nonlinear model problem. Insight gained from the analysis leads to new ux formulas that are stable and that preserve the compactness of the method. Although developed for a model equation, the ux formulas are applicable to systems such as the Euler equations. This article presents the analysis for methods with a degree up to 5. The analysis is accompanied by supporting numerical experiments using Burgers' equation and the Euler equations. Introduction The discontinuous Galerkin method is being developed as a means for obtaining a high-order shock capturing-capability on unstructured meshes. This capability is an important step toward achieving an e cient and robust method for aeroacoustic applications. The objective of this work is to determine whether a local eigenvalue analysis can predict the instabilities that have been previously observed, and to use insight gained from the analysis to suggests ways to eliminate the instability. In reference 1, the discontinuous Galerkin method was formulated in a quadrature-free form that reduced both the computational e ort and storage requirements. Reference 2 described the implementation of boundary conditions, including the treatment of curved walls and nonre ecting boundary conditions. In these works, the method was described in detail, and numerical results for scalar advection and for the Euler equations were shown to demonstrate the accuracy and robustness of the method. These studies showed numerically that piecewise linear and piecewise quadratic discontinuCopyright c 1997 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. *Senior member, AIAA ous Galerkin methods were stable without the use of either limiters or added dissipation when applied to the nonlinear Burgers' equation for a shocked case. However, higher order methods diverged immediately after shock formation. Also in reference 1, it was observed in numerical test that the quadratic form was stable only if the ux integral term was evaluated exactly. The discontinuous Galerkin has a number of fundamental properties essential to any robust shock capturing method. In a series of papers, Cockburn and Shu and Cockburn et al. 4; 5 discussed the discontinuous Galerkin method with the use of approximate Riemann solvers, limiters, and totalvariation-diminishing (TVD) Runge-Kutta time discretizations for nonlinear hyperbolic problems. Reference 5 presents the design of a limiter that applies to general triangulations, maintains a high order of accuracy in smooth regions, and guarantees maximum norm stability. Jiang and Shu also proved that the discontinuous Galerkin method satis es a local cell entropy inequality for the square entropy (U ) = U, for arbitrary triangulations in any space dimension, and for any order of accuracy. This proof trivially implies the L2 stability of the method for nonlinear shocked problems in the scalar case. Flux limiting has been demonstrated as a means for stabilizing the shocked case; 7 however, this approach tends to reduce the formal accuracy and to diminish the compactness of the method. This work presents a local eigenvalue analysis that predicts the stability, or instability, of the method for the Burgers' equation when a shock is present. The analysis is possible only because of the compact nature of the discontinuous Galerkin method. The analysis also provides physical insight into possible causes of the instability and leads to ux formulas that eliminate the instability. The rst section describes the discontinuous Galerkin method applied to the nonlinear Burgers' equation and provides the rationale for the analysis. The second section describes in detail the analysis and the conclusions drawn from it. The third sec1 https://ntrs.nasa.gov/search.jsp?R=20040105535 2018-04-02T23:48:27+00:00Z