Modeling and Computing of Fractional Convection Equation

Abstract

In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank–Nicolson method and the explicit Lax–Wendroff method in time, and the second-order numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efficiency of the derived methods are verified by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.