Explicit Discontinuous Galerkin Methods for Linear Hyperbolic Problems
Master of Arts
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 increase in 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 method in a way that retains the same formal order of accuracy, but reduces the stiffness. The second, optimal Runge-Kutta methods, adds additional stages to Runge-Kutta methods and modifies them to customize their stability region to the problem. Three operator modification methods are analyzed analytically and numerically, the mapping technique of Kosloff/Tal-Ezer the covolume filtering technique of Warburton/Hagstrom , and the flux filtering technique of Chalmers, et al. . The covolume filtering 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 thesis considers five top performing methods from the literature on hyperbolic problems and applies them to an unmodified method, a flux filtered method, and a covolume filtered method. Gains of up to 80\% are seen for covolume filtered solutions applied with optimal Runge-Kutta methods, showing the potential for efficient high order solutions of unsteady systems.
Discontinuous Galerkin; Time-stepping; Mathematics