A dynamically consistent exponential scheme to solve some advection-reaction equations with Riesz anomalous diffusion

This work is motivated by a generalization of the well-known one-dimensional Burgers-Fisher equation, considering a Riesz anomalous diffusion. Initial-boundary conditions which are positive and bounded are imposed on a closed and bounded spatial domain. In this manuscript we propose a finite-difference method to approximate the positive and bounded solutions of the anomalous model. The methodology is an exponential and explicit technique which is based on the use of fractional centered differences. The properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite-difference method, as well as the capability of the technique to preserve the positivity, the boundedness and the monotonicity of the approximations. Additionally, we provide some a priori bounds for the numerical solutions and a consistent analysis of the scheme. Some illustrative simulations show that the method is able to preserve the positivity, the boundedness and the monotonicity of the numerical approximations. (C) 2020 Elsevier B.V. All rights reserved.