Unconditionally Dynamically Consistent Numerical Methods with Operator-Splitting for a Reaction-Diffusion Equation of Huxley’s Type

The efficiency of various numerical methods for solving Huxley’s equation—which includes a diffusion term and a nonlinear reaction term—is investigated. Conventional explicit finite difference algorithms often suffer from severe stability limitations and can yield unphysical concentration values. In this study, we collect a range of stable, explicit time integration methods of first to fourth order, originally developed for the diffusion equation, and design treatments of the nonlinear term which ensure that the solution remains within the physically meaningful unit interval. This property, called dynamical consistency, is analytically proven and implies unconditional stability. In addition to this, the most effective ones are identified from the large number of constructed method combinations. We conduct systematic tests in one and two spatial dimensions, also evaluating computational efficiency in terms of CPU time. Our results show that higher-order schemes are not always the most efficient: in certain parameter regimes, second-order methods can outperform their higher-order counterparts.