A dynamically consistent method to solve nonlinear multidimensional advection-reaction equations with fractional diffusion

This work is motivated by an extension of both the Burgers-Fisher and the Burgers-Huxley equations in multiple dimensions, considering Riesz fractional diffusion. Initial-boundary conditions which are positive and bounded are imposed on a closed and bounded set, and a finite-difference method is proposed to approximate the solutions of the fractional model. The methodology is a linear and implicit technique which is based on fractional centered differences. We show in this manuscript that the method can be expressed in vector form using a Minkowski matrix under suitable conditions. The main properties of Minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite-difference method, as well as the capability of the technique to preserve the positivity and the boundedness. Additionally we show that the method is a consistent technique which is stable and convergent, with first order of convergence in time and second order in space. Some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations.