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High-order Linearly Implicit Two-step Peer - Finite Element Methodsfor Time-dependent Pdes

机译：随时间变化的Pdes的高阶线性隐式两步Peer-有限元方法

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摘要

Linearly-implicit two-step peer methods are successfully applied in the numerical solution of ordinary differential and differential-algebraic equations. One of their strengths is that even high-order methods do not show order reduction in computations for stiff problems. With this property, peer methods commend themselves as time-stepping schemes in finite element calculations for time-dependent partial differential equations (PDEs).rnWe have included a class of linearly-implicit two-step peer methods in the finite element software Kardos. There PDEs are solved following the Rothe method, i.e. first discretised in time, leading to linear elliptic problems in each stage of the peer method. We describe the construction of the methods and how they fit into the finite element framework. We also discuss the starting procedure of the two-step scheme and questions of local temporal error control.rnThe implementation is tested for two-step peer methods of orders three to five on a selection of PDE test problems on fixed spatial grids. No order reduction is observed and the two-step methods are more efficient, at least competitive, in comparison with the linearly implicit one-step methods provided in Kardos.
机译：线性隐式两步对等方法已成功地应用于常微分方程和微分代数方程的数值解中。它们的优点之一是，即使是高阶方法也不会在针对刚性问题的计算中显示出阶数减少。凭借这种特性，对等方法在时间相关的偏微分方程（PDE）的有限元计算中被誉为时间步长方案。在有限元软件Kardos中，我们包括了一类线性隐式两步对等方法。遵循Rothe方法求解PDE，即首先及时离散化，从而在对等方法的每个阶段导致线性椭圆问题。我们描述了方法的构造以及它们如何适合有限元框架。我们还讨论了两步法的启动过程以及局部时间误差控制的问题。在固定空间网格上选择PDE测试问题时，对三步法的两步对等方法的实现进行了测试。与Kardos中提供的线性隐式单步方法相比，没有观察到阶数减少，并且两步方法更有效，至少具有竞争力。

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