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

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• 作者：Alex Kanevsky ; Mark H. Carpenter ; David Gottlieb ; Jan S. Hesthaven
• 关键词：High-order ; Discontinuous Galerkin finite element method (DGFEM) ; Implicit– ; explicit (IMEX) method ; Navier– ; Stokes equations
• 刊名：Journal of Computational Physics
• 出版年：2007
• 出版时间：10 August 2007
• 年：2007
• 卷：225
• 期：2
• 页码：1753-1781
• 全文大小：1844 K

文摘

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.