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Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

Author

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  • Chih-Yu Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
    Department of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • Cheng-Yu Ku

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
    Department of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • Li-Dan Hong

    (Department of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • Shih-Meng Hsu

    (Department of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

Abstract

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

Suggested Citation

  • Chih-Yu Liu & Cheng-Yu Ku & Li-Dan Hong & Shih-Meng Hsu, 2021. "Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs," Mathematics, MDPI, vol. 9(13), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1535-:d:586017
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    References listed on IDEAS

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    1. Biazar, Jafar & Hosami, Mohammad, 2017. "An interval for the shape parameter in radial basis function approximation," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 131-149.
    2. Jingang Xiong & Jiancong Wen & Yan-Cheng Liu, 2020. "Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations," Mathematics, MDPI, vol. 8(8), pages 1-16, July.
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