Volume 20, Issue 2
Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

Commun. Comput. Phys., 20 (2016), pp. 512-533.

Published online: 2018-04

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• Abstract

This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

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@Article{CiCP-20-512, author = {}, title = {Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {2}, pages = {512--533}, abstract = {

This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.060915.301215a}, url = {http://global-sci.org/intro/article_detail/cicp/11162.html} }
TY - JOUR T1 - Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions JO - Communications in Computational Physics VL - 2 SP - 512 EP - 533 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.060915.301215a UR - https://global-sci.org/intro/article_detail/cicp/11162.html KW - AB -

This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

Ji Lin, C. S. Chen & Chein-Shan Liu. (2020). Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions. Communications in Computational Physics. 20 (2). 512-533. doi:10.4208/cicp.060915.301215a
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