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

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.