On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms
On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms
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应用于具有刚性反应项的一维标量双曲方程的 IMEX 格式的稳定性问题
DOI：
10.1090/s0025-5718-2011-02463-4
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
A. Martínez
中科院分区：
文献类型：
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作者：
R. Donat;I. Higueras;A. Martínez
将直线法应用于具有源项的双曲偏微分方程会产生一个包含可能具有非常不同的刚度属性的项的 ODE 系统。在这种情况下，隐式-显式龙格-库塔 (IMEX-RK) 方案作为高阶时间积分器特别有用，因为它们允许对对流项进行显式处理，可以使用高度发达的冲击捕获技术对对流项进行离散化，同时对源项进行隐式处理，这对于稳定性是必要的。受到 LeVeque 和 Yee 在 [J.计算。物理。 86 (1990)]，在本文中，我们研究了将某些不变区域保留为弱稳定性约束。对于本文考虑的源项类别，单位区间是模型平衡律的不变区域。在本文的第一部分中，我们考虑一阶时间离散化，它是高阶 IMEX-RK 方案的基本构建块，并研究保证 [0, 1] 也是数值方案的不变区域的条件。在本文的第二部分，我们研究了确保高阶 IMEX 方案保留此属性的条件。
The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons. Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEX-RK schemes, and study the conditions that guarantee that [0, 1] is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes.