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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes
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The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

机译：在不规则结构域和笛卡尔网格上获得新的数值方法的新数值方法的纳米南边界条件

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摘要

Recently we have developed a new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Poisson equation on irregular domains with the Dirichlet boundary conditions. Its extension to the Neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used with the new approach for 3-D irregular domains. The Neumann boundary conditions are introduced as the known right-hand side into the stencil and do not change the width of the stencil. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1 h-10(-9) h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fourth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations. (C) 2020 ElsevierB.V. All rights reserved.
机译：最近，我们开发了一种新的三维数值方法，用于时间相关的波和热方程，以及具有Dirichlet边界条件的不规则结构域的时间独立泊松方程。在本文中考虑了由于沿着不规则边界存在正常衍生物而导致的Neumann边界条件的延伸。琐碎的笛卡尔网格和简单的27点均匀和非均匀的模板方程与3-D不规则结构域的新方法一起使用。 Neumann边界条件作为已知的右手侧引入模板中，并不会改变模板的宽度。模板方程的系数的计算基于模板方程的局部截断误差的最小化，并产生新技术的精度的最佳顺序。在笛卡尔网格网格的网格点和边界之间的网格点之间非常小的距离（0.1h-10（-9）h，其中H是网格尺寸），边界不会恶化新技术的准确性。在类似的27点模板上，新方法的准确性远高于线性有限元的精度。不规则域的数值结果表明，在相同数量的自由度，新方法比具有更广泛的模板更广泛的模板的高阶（最多四阶）四面体有限元更准确。通过新方法均匀地处理波和热方程。这些等式中的时间衍生的顺序不会影响半分立系统的模板方程的系数。新方法可以直接应用于其他部分微分方程。 （c）2020 elsevierb.v。版权所有。

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