A dimension splitting method for time dependent PDEs on irregular domains. (English) Zbl 1504.65183
Summary: 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65F05 | Direct numerical methods for linear systems and matrix inversion |
| 35R35 | Free boundary problems for PDEs |
Keywords:
time dependent PDEs; time splitting method; Cartesian grid; ADI method; LOD method; kernel-free boundary integral methodReferences:
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