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Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations

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Abstract

We present a family of novel explicit numerical methods for the diffusion or heat equation with Fisher, Huxley and Nagumo-type reaction terms. After discretizing the space variables as in conventional method of lines, our methods do not apply a finite difference approximation for the time derivatives, they instead combine constant- linear- and quadratic-neighbour approximations, which decouple the ordinary differential equations. In the obtained methods, the time step size appears in exponential form in the final expression with negative coefficients. In the case of the pure heat equation, the new values of the variable are convex combinations of the old values, which guarantees unconditional positivity and stability. We analytically prove that the convergence of the methods is fourth order in the time step size for linear ODE systems. We also prove that the concentration values in the case of Fisher’s and Nagumo’s equations lie within the unit interval regardless of the time step size. We construct an adaptive time step size time integrator with an extremely cheap embedded error control method. Several numerical examples are provided to demonstrate that the proposed methods work for nonlinear equations in stiff cases as well. According to the comparisons with other solvers, the new methods can have a significant advantage.

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Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Funding

This research was supported by the EU and the Hungarian State, co-financed by the ERDF in the framework of the GINOP-2.3.4-15-2016-00004 project for the first author.

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  1. Institute of Physics and Electrical Engineering, University of Miskolc, Miskolc, Hungary

    Endre Kovács, János Majár & Mahmoud Saleh

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  1. Endre Kovács
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Contributions

Conceptualization, design, methodology, supervision and the one-dimensional numerical investigation are due to E.K. Analytical proofs were performed by J.M. The research on the adaptive step size controller and the related investigation were done by M.S. The first draft of the manuscript was written by E.K. and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Endre Kovács.

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Kovács, E., Majár, J. & Saleh, M. Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations. J Sci Comput 98, 39 (2024). https://doi.org/10.1007/s10915-023-02426-9

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