A bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion

This work is motivated by a generalization of the well-known Burgers-Fisher and Burgers-Huxley equations in multiple dimensions, considering Riesz fractional diffusion and convection. Initial-boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain. In this manuscript we propose a finite-difference method to approximate the positive and bounded solutions of the fractional model. The methodology is a linear three-steps Crank-Nicolson 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 and the boundedness of the approximations. We show in this work that the method is capable of preserving some of the constant solutions of the continuous model. Additionally, we prove that our technique is a second-order consistent, stable and quadratically convergent scheme. Suitable bounds for the numerical solutions are also derived in this work. Finally, some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations, in agreement with the analytic results proved in this work. Numerical comparisons provided in this work confirm the rate of convergence of the numerical technique.