Dynamically knots setting in meshless method for solving time dependent propagations equation
Dynamically knots setting in meshless method for solving time dependent propagations equation
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DOI：
10.1016/j.cma.2003.12.015
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发表时间：
2004-03
影响因子：
7.2
通讯作者：
Zongmin Wu
中科院分区：
文献类型：
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作者：
Zongmin Wu
为了有效地显示具有有限离散采样数据的函数，我们需要在函数更振荡的地方更多的采样数据，而在函数更平坦的地方更少的采样数据。如果函数恰好是偏微分方程的解，用有限元法求解，则应在奇异点附近构造更细的单元。然而，我们不知道一个函数的振荡甚至是冲击会发生在哪里，这是一个非线性偏微分方程的解。因此，我们不能在这些点附近预设更精细的元素。一个简单的方法是采取非常密集的节点或非常精细的元素无处不在，以保持精度。这种策略耗费了大量的计算时间。另一种思想是根据函数随时间的变化来移动采样点的位置。我们观察到，这种方法可以实现拉格朗日列式有限差分法。然而，拉格朗日公式不具有形状保持和变差减小特性。由于节点的移动会破坏网格的拓扑结构，因此有限元方法也很难实现这种方法。近年来，无网格方法成为数值求解偏微分方程的研究热点。无网格方法不需要任何网格或节点（采样点）的任何结构，因此我们可以自由移动节点来模拟问题。唯一的限制是保持结不重叠。本文是一个测试我们的方法与径向基拟插值的伯格方程。
In order to display a function with some finite discrete sampling data efficiently, we require more sampling data where the function is more oscillatory, and less sampling data where the function is more flat. If the function is happen to be a solution of partial differential equation and is solved by finite elements method, then we should construct finer element near the singularity. However, we do not know where the oscillation or even shocks will happen to a function, which is a solution of non-linear partial differential equation. Therefore we cannot preset the finer elements near such points. A trivial method is taking very dense knots or very fine elements everywhere to keep the accuracy. This strategy cost the computation time. Another idea is to move the position of the sampling points according to the varying of the function with the time. We observed that, this approach could be achieved by Lagrangian formulation for finite difference method. However the Lagrangian formulation does not possess shape preserving and variation diminishing properties. It is difficult to achieve the approach for finite elements methods too, because the moving knots will destroy the topology of the mesh. Recently the meshless method becomes to topic to solve partial differential equation numerically. The meshless method does not require any mesh or any structure of the knots (sampling points); therefore we can move the knots freely to simulate the problem. The only restriction is to keep the knots no overlapping. This paper is a test of our approach for the Burger's equation with radial basis quasi-interpolation.