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A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains

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Abstract

In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of \(O(N \log N)\), where N stands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions.

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Acknowledgements

The research of Long was supported in part by the Natural Science Foundation of Guangxi in China under grant AD20238065, and the key project of Guangxi Provincial Natural Science Foundation of China under grant 2018GXNSFDA050014. The research of Boerman was supported in part by the grant of Research in Mathematics and the Sciences (RIMS) offered by the College of the Sciences and Mathematics at West Chester University of Pennsylvania, USA. The research of Zhao was supported in part by the National Science Foundation (NSF) grant DMS-2110914.

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Authors and Affiliations

  1. Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA, 19383, USA

    Chuan Li & Eric Boerman

  2. Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA

    Yiming Ren & Shan Zhao

  3. Guangxi Key Lab of Human-Machine Interaction and Intelligent Decision, Nanning Normal University, Nanning, 530001, People’s Republic of China

    Guangqing Long

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  1. Chuan Li
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  3. Guangqing Long
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Li, C., Ren, Y., Long, G. et al. A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains. J Sci Comput 95, 49 (2023). https://doi.org/10.1007/s10915-023-02177-7

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