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首页> 外文期刊>SIAM Journal on Scientific Computing >ARBITRARY LAGRANGIAN EULERIAN-TYPE FINITE ELEMENT METHODS FORMULATION FOR PDEs ON TIME-DEPENDENT DOMAINS WITH VANISHING DISCRETE SPACE CONSERVATION LAW
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ARBITRARY LAGRANGIAN EULERIAN-TYPE FINITE ELEMENT METHODS FORMULATION FOR PDEs ON TIME-DEPENDENT DOMAINS WITH VANISHING DISCRETE SPACE CONSERVATION LAW

机译：任意拉格朗日Eulerian型有限元方法在消失离散空间守恒法中对时间依赖域的PDE

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摘要

The aim of this paper is to introduce a finite element formulation within an arbitrary Lagrangian Eulerian (ALE) framework with a vanishing discrete space conservation law (SCL) for differential equations on time-dependent domains. The novelty of the formulation is the method for temporal integration which results in preserving the SCL property and retaining the higher order accuracy at the same time. Once the time derivative is discretized (based on an integration or differentiation formula), the common approach for terms in differential equation which do not involve temporal derivative is classified to be a kind of "time averaging" between time steps. In the spirit of classical approaches, this involves evaluating these terms at several points in time between the current and the previous time step ([t(n), t(n+1)]), and then averaging them in order to provide the satisfaction of discrete SCL. Here, we fully use the polynomial in time form of mapping through which the evolution of the domain is realized-the so-called ALE map-in order to avoid the problems arising due to the moving grids. We give a general recipe on temporal schemes that have to be employed once the discretization for the temporal derivative is chosen. Numerical investigations on stability, accuracy, and convergence are performed and the simulated results are compared with benchmark problems set up by other authors.
机译：本文的目的是在任意拉格朗日欧拉（ALE）框架内引入有限元制剂，其具有消失的离散空间守恒法（SCL），用于在时间依赖域上的微分方程。制剂的新颖性是用于时间集成的方法，从而导致保留SCL性能并同时保留更高阶精度。一旦离散时间衍生物（基于积分或分化公式），则不涉及时间衍生物的差分方程的术语的常见方法被分类为时间步长之间的一种“时间平均”。本着古典方法的精神，这涉及在当前和先前时间步骤之间的几个时间点（[T（n），t（n + 1）]），然后平均它们以提供它们以提供离散SCL的满意度。这里，我们充分利用多项式的映射形式，通过该映射来实现域的演变 - 所谓的ALE图 - 为了避免由于移动网格引起的问题而产生的问题。我们在选择临时衍生物的离散化后，给出了一般性方案的一般配方。对稳定性，准确性和收敛的数值研究，并将模拟结果与其他作者设置的基准问题进行了比较。

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