Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:233

Article

Sanderse, B.^1,2 

[1] Energy Res Ctr Netherlands ECN, Petten, Netherlands

[2] Ctr Wiskunde & Informat, Amsterdam, Netherlands

关键词: Energy conservation;    Time reversibility;    Runge-Kutta method;    Additive Runge-Kutta method;    Incompressible Navier-Stokes equations;    Differential-algebraic equations;    Algebraic stability;    L stability;    Stiffness;   

DOI  :  10.1016/j.jcp.2012.07.039

来源: Elsevier

【 摘 要 】

Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier-Stokes equations. In this paper implicit Runge-Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge-Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge-Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convection-dominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier-Stokes equations. (C) 2012 Elsevier Inc. All rights reserved.

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