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RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

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  • Raeisi, Banafsheh
  • Fardi, Mojtaba
  • Darani, Mohammadreza Ahmadi

Abstract

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

Suggested Citation

  • Raeisi, Banafsheh & Fardi, Mojtaba & Darani, Mohammadreza Ahmadi, 2024. "RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 226(C), pages 152-171.
    • Handle: RePEc:eee:matcom:v:226:y:2024:i:c:p:152-171
    DOI: 10.1016/j.matcom.2024.07.001
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    References listed on IDEAS

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    1. Nikan, O. & Avazzadeh, Z., 2021. "A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    2. Benkhaldoun, Fayssal & Halassi, A. & Ouazar, Driss & Seaid, Mohammed & Taik, Ahmed, 2017. "A stabilized meshless method for time-dependent convection-dominated flow problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 137(C), pages 159-176.
    3. Cavoretto, Roberto & De Rossi, Alessandra & Lancellotti, Sandro & Romaniello, Federico, 2024. "Node-bound communities for partition of unity interpolation on graphs," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    4. Nikan, O. & Avazzadeh, Z., 2022. "A locally stabilized radial basis function partition of unity technique for the sine–Gordon system in nonlinear optics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 394-413.
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