Journal of Scientific Computing

, Volume 60, Issue 2, pp 313–344 | Cite as

Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

  • Ethan J. Kubatko
  • Benjamin A. Yeager
  • David I. Ketcheson
Article

Abstract

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.

Keywords

Discontinuous Galerkin Runge–Kutta Strong-stability-preserving 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ethan J. Kubatko
    • 1
  • Benjamin A. Yeager
    • 1
  • David I. Ketcheson
    • 2
  1. 1.Department of Civil, Environmental, and Geodetic EngineeringThe Ohio State UniversityColumbusUSA
  2. 2.Division of Mathematical and Computer Sciences and Engineering4700 King Abdullah University of Science and TechnologyThuwalSaudi Arabia

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