Abstract
We develop a simple and efficient dimension splitting method for solving time dependent partial differential equations (PDEs) on multiple space dimensional irregular domains using a one dimensional kernel-free boundary integral (KFBI) method. The proposed method extends the alternating direction implicit methods and locally one dimensional methods to more general cases involving complex geometry. The KFBI method is a potential theory based Cartesian grid method, which works as an improvement of conventional boundary integral methods. In the KFBI method, boundary or volume integrals are evaluated by solving equivalent interface problems without using any analytic expression of Green’s functions. The one dimensional interface problems after dimension splitting are solved by finite difference method. The resulting linear systems are tri-diagonal and efficiently solved by the Thomas algorithm. The one dimensional kernel-free boundary integral method is rigorously proved to have second-order convergence rate in the maximum norm. Multiple numerical examples, including different types of PDEs and a free boundary problem, are presented to demonstrate the advantages of the proposed method. Numerical results show that the proposed method is efficient and achieves overall second order accuracy.
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Data availibility
The custom code of the current study is available at https://github.com/zhouhan-sjtu/KFBI-OS. The data generated in this study are available upon reasonable request.
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This work is financially supported by the National Key R & D Program of China, Project Number 2020YFA0712000. It is also partially supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), the National Natural Science Foundation of China (Grant No. DMS-11771290) and the Science Challenge Project of China (Grant No. TZ2016002).
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This work is financially supported by the National Key R &D Program of China, Project Number 2020YFA0712000. It is also partially supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), the National Natural Science Foundation of China (Grant No. DMS-11771290) and the Science Challenge Project of China (Grant No. TZ2016002).
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Zhou, H., Ying, W. A Dimension Splitting Method for Time Dependent PDEs on Irregular Domains. J Sci Comput 94, 20 (2023). https://doi.org/10.1007/s10915-022-02066-5
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DOI: https://doi.org/10.1007/s10915-022-02066-5
Keywords
- Time dependent PDEs
- Time splitting method
- Cartesian grid
- ADI method
- LOD method
- Kernel-free boundary integral method