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A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics

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  • Nikan, O.
  • Avazzadeh, Z.

Abstract

This paper develops a numerical approach for finding the approximate solution of the Sobolev model. This model describes many natural processes, such as thermal conduction for different media and fluid evolution in soils and rocks. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, the time discretization is performed by means of a second-order finite difference procedure. At a second stage, the space discretization is accomplished using the local radial basis function partition of unity collocation method based on the finite difference (LRBF-PUM-FD). The major disadvantage of global techniques is the high computational burden of solving large linear systems. The LRBF-PUM-FD significantly sparsifies the linear system and reduces the computational burden, while simultaneously maintaining a high accuracy level. The time-discrete formulation is studied in terms of the stability and convergence analysis via the energy method. Three examples are illustrated to verify the efficiency and accuracy of the method.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:401:y:2021:i:c:s0096300321001119
    DOI: 10.1016/j.amc.2021.126063
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    References listed on IDEAS

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    1. Su, LingDe, 2019. "A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 232-247.
    2. Zhao, Wei & Hon, Y.C. & Stoll, Martin, 2018. "Numerical simulations of nonlocal phase-field and hyperbolic nonlocal phase-field models via localized radial basis functions-based pseudo-spectral method (LRBF-PSM)," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 514-534.
    3. Zhang, Jiansong & Zhang, Yuezhi & Guo, Hui & Fu, Hongfei, 2019. "A mass-conservative characteristic splitting mixed finite element method for convection-dominated Sobolev equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 180-191.
    4. Cavoretto, Roberto & De Rossi, Alessandra, 2020. "Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    5. Luo, Zhendong & Teng, Fei, 2018. "A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 374-383.
    6. You, Xiangyu & Li, Wei & Chai, Yingbin, 2020. "A truly meshfree method for solving acoustic problems using local weak form and radial basis functions," Applied Mathematics and Computation, Elsevier, vol. 365(C).
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    1. Nikan, O. & Avazzadeh, Z. & Tenreiro Machado, J.A., 2021. "Numerical simulation of a degenerate parabolic problem occurring in the spatial diffusion of biological population," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).

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