Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation
Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation
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DOI：
10.1137/17m1144362
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发表时间：
2017-08
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影响因子：
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通讯作者：
Jingwei Hu;Ruiwen Shu;Xiangxiong Zhang
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文献类型：
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作者：
Jingwei Hu;Ruiwen Shu;Xiangxiong Zhang
我们为刚性BGK动力学方程建立了一类二阶隐显格式。该方法是渐近保持的（可以捕获欧拉极限，而不需要数值求解小Knudsen数）以及正保持的-一个现有的任何二阶或高阶IMEX方案都不具有的特征。该方法基于常用的IMEX龙格-库塔框架加上利用BGK算子特殊结构的关键校正步骤。形式分析证明了该方法的性质，并得到了各种数值结果的支持。此外，我们还证明了当适当的空间离散化耦合时，该方法满足熵衰减特性。此外，我们讨论了该方法在某些双曲松弛系统中的推广，并提供了一种将该方法推广到三阶的策略。
We develop a family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation. The method is asymptotic-preserving (can capture the Euler limit without numerically resolving the small Knudsen number) as well as positivity-preserving --- a feature that is not possessed by any of the existing second or high order IMEX schemes. The method is based on the usual IMEX Runge-Kutta framework plus a key correction step utilizing the special structure of the BGK operator. Formal analysis is presented to demonstrate the property of the method and is supported by various numerical results. Moreover, we show that the method satisfies an entropy-decay property when coupled with suitable spatial discretizations. Additionally, we discuss the generalization of the method to some hyperbolic relaxation system and provide a strategy to extend the method to third order.