The upwind PPM scheme and analysis for solving two-sided space-fractional advection-diffusion equations in three dimension

In this paper, we first analyze the mass conservative characteristic finite difference method for solving three dimensional two-sided space-fractional advection-diffusion equation by combining the operator splitting technique. Along each direction, by using the upwind piecewise parabola method which meets mass conservation, we compute firstly the predecessor numerical solutions. Secondly, we solve the equations with the shifted Grunwald-Letnikov approximations by the splitting implicit characteristic difference scheme. By some auxiliary lemmas, we prove strictly that our scheme is unconditionally stable and is of first-order convergence in space and time. Numerical experiments are used to verify our theoretical analysis.