Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation

Husniddin Khayrullaev, Issa Omle and Endre Kovács ()
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Husniddin Khayrullaev: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Issa Omle: 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

Mathematics, 2025, vol. 13, issue 2, 1-38

Abstract: Four explicit numerical schemes are collected, which are stable and efficient for the diffusion equation. Using these diffusion solvers, several new methods are constructed for the nonlinear Huxley’s equation. Then, based on many successive numerical case studies in one and two space dimensions, the least performing methods are gradually dropped out to keep only the best ones. During the tests, not only one but all the relevant time step sizes are considered, and for them, running-time measurements are performed. A major aspect is computational efficiency, which means that an acceptable solution is produced in the shortest possible time. Parameter sweeps are executed for the coefficient of the nonlinear term, the stiffness ratio, and the length of the examined time interval as well. We obtained that usually, the leapfrog–hopscotch method with Strang-type operator-splitting is the most efficient and reliable, but the method based on the Dufort–Frankel scheme can also be very efficient.

Keywords: nonlinear PDEs; diffusion–reaction equations; Huxley’s equation; explicit numerical methods; stiff equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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