Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit
Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit
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DOI：
10.1063/1.3637861
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发表时间：
2011-09
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影响因子：
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通讯作者：
S. Boscarino;L. Pareschi;G. Russo
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文献类型：
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作者：
S. Boscarino;L. Pareschi;G. Russo
我们考虑具有刚性松弛的双曲组的隐式-显式(IMEX)Runge-Kutta(R-K)格式。在这样的区域中，系统向对流扩散方程松弛。本文的第一个目的是证明传统的分块IMEX R-K格式将在不需要修改原系统的情况下松弛到极限方程的显式格式。当然，在极限下得到的显式格式在时间步长上受到经典抛物线稳定性的限制。本文的主要目的是在IMEX R-K格式的基础上，将扩散极限放宽为对流扩散方程的IMEX R-K格式，其中扩散项被隐式处理。这是通过对问题的新的重新表述以及随后对其应用IMEX R-K格式来实现的。对改写后的问题的分析表明，对于极限方程，这些格式简化为IMEX R-K格式。
We consider implicit-explicit (IMEX) Runge--Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such a regime the system relaxes towards a convection-diffusion equation. The first objective of this paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of this paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by a novel reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis of such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, unde...