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Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

Author

Listed:
  • Humam Kareem Jalghaf

    (Department of Fluid and Heat Engineering, University of Miskolc, 3515 Miskolc, Hungary
    Mechanical Engineering Department, University of Technology, Baghdad 10066, Iraq)

  • Endre Kovács

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

  • János Majár

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

  • Ádám Nagy

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

  • Ali Habeeb Askar

    (Department of Fluid and Heat Engineering, University of Miskolc, 3515 Miskolc, Hungary
    Mechanical Engineering Department, University of Technology, Baghdad 10066, Iraq)

Abstract

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3308-:d:705942
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    Citations

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    Cited by:

    1. Issa Omle & Ali Habeeb Askar & Endre Kovács & Betti Bolló, 2023. "Comparison of the Performance of New and Traditional Numerical Methods for Long-Term Simulations of Heat Transfer in Walls with Thermal Bridges," Energies, MDPI, vol. 16(12), pages 1-27, June.
    2. 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.
    3. Imre Ferenc Barna & László Mátyás, 2022. "Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations," Mathematics, MDPI, vol. 10(18), pages 1-17, September.

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