Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes

  • Authors:

  • Alex Kanevsky;Mark H. Carpenter;David Gottlieb;Jan S. Hesthaven

  • Affiliations:

  • Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA;Aeronautics and Aeroacoustic Methods Branch, NASA Langley Research Center, Hampton, VA 23681-0001, USA;Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA;Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

Despite the popularity of high-order explicit Runge-Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems. The IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta method (ESDIRK). Furthermore, we apply adaptive time-step controllers based on the embedded temporal error predictors. We demonstrate in a number of numerical test problems that IMEX methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid-induced stiffness.