Numerical Computation of the Wave Speed for Hyperbolic Reaction-Diffusion Equations

This contribution focuses on numerical approximation of propagation speed in (hyperbolic) modeling of reactive-diffusive phenomena. Firstly, we start from the standard derivation of reaction-diffusion equation as a consequence of the coupling of the continuity equation with the Fourier’s law. Next, we propose to attack the problem by means of hyperbolic models, a crucial example obtained by replacing the (static) Fourier equality with the (dynamic) Maxwell–Cattaneo identity. Then, we concentrate on planar fronts for scalar reaction-diffusion equation focusing on bistable nonlinearities. Thereafter, we turn to the third part which is devoted to numerical schemes. To begin with, we propose the phase plane algorithm, providing a direct approximation of the propagation speed of the traveling wave by means of a standard shooting argument. Once validated such scheme—by comparison with the case of reaction-diffusion equation with damping for which an exact formula is explicitly known—we focus on the relaxation case, assuming as exact value the one computed by the phase-plane algorithm. For such case, we consider a kinetic IMEX algorithm for the original hyperbolic system where only the linear terms of the operator are discretized implicitly and we propose two algorithms to approximate the propagation speed exploring their capability to predict the correct value. The first one, named scout &spot algorithm, is based on the determination of the slope of some appropriate level curve; the second one, based on a discretization of the LeVeque-Yee formula, approximates the velocity value by means of an integral average. A widely extended version of this manuscript is contained in [11].