Implicit-explicit schemes for flow equations with stiff source terms
Implicit-explicit schemes for flow equations with stiff source terms
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具有刚性源项的流动方程的隐式-显式格式
DOI：
10.1016/j.cam.2010.08.015
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Siddhartha Mishra
中科院分区：
文献类型：
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作者：
M. Svärd;Siddhartha Mishra
本文设计了具有刚性源项的守恒律的稳定、精确的数值格式。一个主要的例子和我们研究的主要动机是气体动力学的反应欧拉方程。此外，我们考虑了广泛研究的标量模型方程。我们为这些方程设计了一步式IMEX（隐式-显式）方案，显式地处理对流项，隐式地处理源项。对于非线性标量方程，我们使用了一种新颖的初始数据选择，并得到了正确的传播速度，即使在初始数据稀疏的困难情况下。对于反应性欧拉方程，为了在欠分解网格上得到正确的波速，我们适当地选择了数值扩散。我们证明了我们的隐显格式在标量情况下是收敛的，并给出了大量的数值实验来验证我们的格式在标量情况和反应性欧拉方程情况下的有效性。此外，我们还讨论了在设计方案时必须考虑的反应性欧拉方程和标量模型方程之间的基本区别。
In this paper, we design stable and accurate numerical schemes for conservation laws with stiff source terms. A prime example and the main motivation for our study is the reactive Euler equations of gas dynamics. Furthermore, we consider widely studied scalar model equations. We device one-step IMEX (implicit–explicit) schemes for these equations that treats the convection terms explicitly and the source terms implicitly. For the non-linear scalar equation, we use a novel choice of initial data for the resulting Newton solver and obtain correct propagation speeds, even in the difficult case of rarefaction initial data. For the reactive Euler equations, we choose the numerical diffusion suitably in order to obtain correct wave speeds on under-resolved meshes. We prove that our implicit–explicit scheme converges in the scalar case and present a large number of numerical experiments to validate our scheme in both the scalar case as well as the case of reactive Euler equations. Furthermore, we discuss fundamental differences between the reactive Euler equations and the scalar model equation that must be accounted for when designing a scheme.