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New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient

Author

Listed:
  • Mahmoud Saleh

    (Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary)

  • Endre Kovács

    (Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary)

  • Imre Ferenc Barna

    (Wigner Research Center for Physics, 1051 Budapest, Hungary)

  • László Mátyás

    (Department of Bioengineering, Sapientia Hungarian University of Transylvania, 530104 Miercurea Ciuc, Romania)

Abstract

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Suggested Citation

  • Mahmoud Saleh & Endre Kovács & Imre Ferenc Barna & László Mátyás, 2022. "New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient," Mathematics, MDPI, vol. 10(15), pages 1-26, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2813-:d:883168
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    References listed on IDEAS

    as
    1. Humam Kareem Jalghaf & Endre Kovács & János Majár & Ádám Nagy & Ali Habeeb Askar, 2021. "Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations," Mathematics, MDPI, vol. 9(24), pages 1-21, December.
    2. Somayeh Pourghanbar & Jalil Manafian & Mojtaba Ranjbar & Aynura Aliyeva & Yusif S. Gasimov, 2020. "An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, November.
    3. Ndivhuwo Ndou & Phumlani Dlamini & Byron Alexander Jacobs, 2022. "Enhanced Unconditionally Positive Finite Difference Method for Advection–Diffusion–Reaction Equations," Mathematics, MDPI, vol. 10(15), pages 1-18, July.
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    Cited by:

    1. Yi Ji & Yufeng Xing, 2023. "Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation," Mathematics, MDPI, vol. 11(3), pages 1-36, January.

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