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

• Published in: Journal of computational physics Vol. 366; pp. 71 - 88
• Main Author: Macías-Díaz, J.E.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.08.2018; Elsevier Science Ltd
• Subjects: Advection–diffusion–reaction partial differential equations; Computational physics; Computer simulation; Convergence; Differential equations; Finite difference method; Fractional centered differences; Implicit finite-difference scheme; Mathematical models; Matrix; Matrix algebra; Matrix methods; Multi-consistent numerical method; Nonlinear equations; Partial differential equations; Riesz space-fractional equation; Stability and convergence analyses; Riesz space-fractional equation; Fractional centered differences; Implicit finite-difference scheme; Stability and convergence analyses; Advection–diffusion–reaction partial differential equations; Multi-consistent numerical method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.03.047

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. •An efficient method for fractional-diffusion equations diffusion is introduced.•The method is a dynamically consistent linear technique.•The method preserves positivity, boundedness and monotonicity.•The consistency, stability and convergence are rigorously established.