Finite Difference Methods For Two Kinds Of Nonstationary Fractional Convection-Diffusion Equations

In this paper, we consider two kinds of fractional convection-diffusion equations, one is space-fractional convection-diffusion equation and the other one is time-space fractional convection-diffusion equation. Based on the shifted Grunwald formula, we use the weighted average finite difference method to approximate the spatial Riemann-Liouville fractional derivatives in the first equation, study its stability by eigenvalue analysis, and we obtain the error estimate is O(τ+h). A high order approximation for the temporal derivative is used for the second equation, the stability is given by the technique of the maximum norm analysis, with the convergence order O(τ2-max{γ1,γ2), here γ1,γ2are the orders of the two time Caputo fractional derivatives, respectively. Numerical examples are presented to demonstrate the theoretical results.