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

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.