Explicit Discontinuous Galerkin Methods for Linear Hyperbolic Problems

[摘要] Discontinuous Galerkin methods have many features which make them a natural candidate for the solution of hyperbolic problems. One feature is flexibility with the order of approximation; a user with knowledge of the solution's regularity can increase the spatial order of approximation by increasing the polynomial order of the discontinuous Galerkin method. A marked increase in time-stepping difficulty, known as stiffness, often accompanies this increasein spatial order however. This thesis analyzes two techniques for reducing the impact of this stiffness on total time of simulation. The first, operator modification, directly modifies the high order methodin a way that retains the same formal order of accuracy, but reduces the stiffness. The second, optimalRunge-Kutta methods, adds additional stages to Runge-Kutta methods and modifies them tocustomize their stability region to the problem. Threeoperator modification methods are analyzed analytically and numerically, the mapping technique of Kosloff/Tal-Ezerthe covolume filtering technique of Warburton/Hagstrom , andthe flux filtering technique of Chalmers, et al. . The covolumefiltering and flux filtering techniques outperform mapping in that they negligibly impact accuracy but yield a reasonable improvement in efficiency. For optimal Runge-Kutta methods this thesisconsiders five top performing methods from the literature on hyperbolic problems and applies them to an unmodified method, a flux filteredmethod, and a covolume filtered method. Gains of up to 80\% are seen for covolume filtered solutions appliedwith optimal Runge-Kutta methods, showing the potential for efficient high order solutions of unsteady systems.

[发布机构] Rice University

[效力级别] Galerkin