The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

Junli Zhang, Chenchen Yang, Hui Zheng, Chia-Ming Fan and Ming-Fu Fu

Mathematics and Computers in Simulation (MATCOM), 2022, vol. 200, issue C, 504-524

Abstract: In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.

Keywords: Localized method of fundamental solutions; Meshless method; Recursive composite multiple reciprocity method; Multi-dimensional boundary value problems; Inhomogeneous partial differential equations (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:200:y:2022:i:c:p:504-524

DOI: 10.1016/j.matcom.2022.04.024

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