Volumetric variational principles for a class of partial differential equations defined on surfaces and curves: In memory of Heinz-Otto Kreiss
Volumetric variational principles for a class of partial differential equations defined on surfaces and curves: In memory of Heinz-Otto Kreiss
复制标题
在曲面和曲线上定义的一类偏微分方程的体积变分原理：纪念 Heinz-Otto Kreiss
DOI：
10.1007/s40687-018-0137-1
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发表时间：
2018
影响因子：
1.2
通讯作者：
Tsai, Richard
中科院分区：
文献类型：
--
作者：
Chu, Jay;Tsai, Richard
在本文中，我们提出了一种简单的数值算法，用于定义在封闭光滑表面（或曲线）上的偏微分方程。特别地，我们考虑了源自曲面上定义的变分原理的偏微分方程；这些方程包括拉普拉斯-贝尔特拉米方程和表面波方程。该方法是系统地表述变分积分的扩展，并推导扩展问题的欧拉-拉格朗日方程，包括可以在均匀笛卡尔网格或自适应网格上容易离散的边界条件。在我们的方法中，曲面由距离函数或最近点映射隐式定义。由于这些扩展并不是唯一的，因此我们将研究一类简单扩展如何影响生成的pde。特别地，我们将表面偏微分方程简化为在周期带和相应边界条件上定义的问题的模型，并使用经典的傅里叶和拉普拉斯变换方法来研究由此产生的问题的适定性。对于椭圆型和抛物型问题，我们的边界闭包大多给出了求解非线性曲面偏微分方程的稳定算法。对于双曲型问题，所提出的边界闭包通常是不稳定的，但通过添加高阶正则化项或周期性但不频繁地“重新初始化”计算解，可以很容易地控制这种不稳定性。给出了各典型曲面偏微分方程的数值算例。
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace–Beltrami equations and surface wave equations. The approach is to systematically formulate extensions of the variational integrals and derive the Euler–Lagrange equations of the extended problem, including the boundary conditions that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher-order regularization term or by periodically but infrequently “reinitializing” the computed solutions. Some numerical examples for each representative surface PDEs are presented.