The Second Order Modified Upwind PPM Scheme for Solving Space-Fractional Advection-Diffusion Equations in Three Dimensions

In this paper, we propose the second-order modified upwind conservative characteristic difference method for solving space-fractional advection-diffusion equations by combining the operator splitting. Firstly, the intermediate numerical solution is computed by using the piecewise parabolic method (PPM) that satisfies mass conservation and preserves the modified second order in space. Secondly, the numerical solutions are obtained by the weighted shifted Grunwald-Letnikov difference scheme. By some auxiliary lemmas, we prove strictly that our scheme is stable under the condition Delta t=O(h2) in L2-norm, and are of second-order convergence in space and first-order convergence in time. Numerical experiments are given to verify the theoretical results.