On the stability of explicit finite difference methods for advection–diffusion equations
On the stability of explicit finite difference methods for advection–diffusion equations
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DOI：
10.1002/num.22897
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发表时间：
2020-06
影响因子：
3.9
通讯作者：
Xianyi Zeng;M. K. Hasan
中科院分区：
文献类型：
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作者：
Xianyi Zeng;M. K. Hasan
本文研究对流扩散方程任意阶精度显式差分离散的稳定性。分析首先集中在常微分方程系统的稳定性，通过离散ADE在空间中，然后扩展到完全离散的方法结合显式龙格库塔方法。特别地，我们证明了只要时间积分器至少是一阶精度，ADE的所有稳定的半离散化都会导致条件稳定的完全离散化方法，而对流方程的高阶空间离散化如果时间阶太低则不能产生稳定的方法。在文章的后半部分，分析和稳定性的结果被扩展到一个部分耗散波系统，它作为一个模型，在许多流体力学的应用，将粘性应力的动量方程，但没有热耗散的能量方程中的共同做法。最后，通过数值算例验证了主要的理论预测.
In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.