Moving Mesh Generation Using the Parabolic Monge--Amp[e-grave]re Equation
Moving Mesh Generation Using the Parabolic Monge--Amp[e-grave]re Equation
复制标题
使用抛物线 Monge--Amp[e-grave]re 方程生成移动网格
DOI：
10.1137/080716773
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
J. F. Williams
中科院分区：
文献类型：
--
作者：
C. Budd;J. F. Williams
本文提出了一种新的运动网格生成方法，该方法适用于若干空间维度的偏微分方程数值解。该网格由满足适当的非线性抛物型偏微分方程的（标量）网格势函数的梯度得到。该方法提出了一种基于最优运输思想和（时变）标量监测函数的均分原理（已成功应用于一维移动网格方法）的r自适应新技术。对该方法进行了详细的分析，研究了网格的收敛性、规律性和稳定性。此外，该方法易于编程和实现，只需要在任意维度上求解一个简单的标量时间相关方程，并具有沿边界自动处理的自适应性。在这项工作的背景下，我们讨论了三种预先存在的方法。举例说明了监测功能是预先规定的，或由偏微分方程的解给出。
This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing $r$-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation.