A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems
A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems
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DOI：
10.1080/10618562.2011.575368
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发表时间：
2011-01-01
影响因子：
1.3
通讯作者：
Sherwin, Spencer J.
中科院分区：
文献类型：
--
作者：
Vos, Peter E. J.;Eskilsson, Claes;Sherwin, Spencer J.
时间步进算法及其实现是瞬态偏微分方程 (PDE) 求解中的关键组成部分。在本文中，我们提出了一个通用框架 - 无论是在算法还是实现方面 - 允许在各种显式、隐式和隐式显式 (IMEX) 时间步进方法之间几乎无缝切换。我们特别强调如何纳入与时间相关的边界条件，这是一个超越经典 ODE 理论的问题，但它在计算流体动力学中出现的偏微分方程的时间步长中发挥着重要作用。我们的算法基于 J.C. Butcher 的通用线性方法的统一概念，我们对其进行了扩展以适应工程实践中经常使用的 IMEX 方案系列。在本文中，我们讨论了设计注意事项并提出了面向对象的实现。最后，我们通过应用于模型问题以及更复杂的流体问题来说明该框架的使用。
Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework - both in terms of algorithms and implementations - that allows an almost seamless switch between various explicit, implicit and implicit-explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.