Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods
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Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.
KeywordsDiscontinuous Galerkin Runge–Kutta Strong-stability-preserving
The first and second author acknowledge support by National Science Foundation Grants DMS-0915118 and DMS-1217218.
- 12.Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pp. 95–110. Springer, Berlin (2008). http://stanford.edu/boyd/~graph_dcp.html
- 13.Grant, M., Boyd, S.: CVX: Matlab Software for Disciplined Convex Programming, version 2.0 beta (2012). http://cvxr.com/cvx
- 23.Parsani, M., Ketcheson, D.I.: Design of optimal explicit linearly stable strong stability preserving Runge–Kutta schemes for the spectral difference method. In : International Conference on Spectral and High Order Methods (2012)Google Scholar
- 25.Reddy, S.C., Trefethen, L.N.: Stability of the method of lines. Numerische Mathematik 62, 235–267 (1992)Google Scholar