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A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Author

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  • Endre Kovács

    (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)

  • Mahmoud Saleh

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

Abstract

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

Suggested Citation

  • Endre Kovács & Ádám Nagy & Mahmoud Saleh, 2021. "A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation," Mathematics, MDPI, vol. 9(18), pages 1-22, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2284-:d:637118
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    References listed on IDEAS

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    1. G. F. Sun & G. R. Liu & M. Li, 2015. "An Efficient Explicit Finite-Difference Scheme for Simulating Coupled Biomass Growth on Nutritive Substrates," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-17, January.
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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.

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