Is Pollution Effect of Finite Difference Schemes Avoidable for Multi-Dimensional Helmholtz Equations with High Wave Numbers?
Is Pollution Effect of Finite Difference Schemes Avoidable for Multi-Dimensional Helmholtz Equations with High Wave Numbers?
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DOI：
10.4208/cicp.oa-2016-0057
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发表时间：
2017-02
影响因子：
3.7
通讯作者：
Kun Wang;Y. Wong
中科院分区：
文献类型：
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作者：
Kun Wang;Y. Wong
本文提出了一种用分离变量法求解二维亥姆霍兹方程的方法。然后通过解决一系列一维问题得到一系列解决方案来计算解决方案，其中一维解决方案使用无污染差分格式计算。此外，构造了无污染的数值积分公式来处理非齐次一维问题中由于强迫项引起的积分。因此，计算出的解不会受到污染的影响。该方法的另一个吸引人的特点是直接法可以有效地求解一维亥姆霍兹方程数值离散后的三对角矩阵。该方法已被用于计算模拟开放大腔和矩形波导电磁散射的二维亥姆霍兹解。AMS学科分类：65N06、65N15、65N22
This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide. AMS subject classifications: 65N06, 65N15, 65N22