The Conservative Characteristic Difference Method and Analysis for Solving Two-Sided Space-Fractional Advection-Diffusion Equations

In this paper, we propose and analyze the mass-conservative characteristic finite difference method for solving two-sided space-fractional advection-diffusion equation. The predecessor numerical solutions are firstly computed by using the piecewise parabola method (PPM) which preserves the local mass conservation. Then, we solve the equations with the shifted Grünwald-Letnikov approximations by the splitting implicit characteristic difference scheme. By some auxiliary lemmas, we prove strictly that our schemes I and II are stable under the condition Δ t = O (Δ x 2 ) based on the choice of the weight coefficient in L 2 -norm, respectively. Their error estimates are given and we prove our scheme I is of first-order and scheme II is second-order convergence in space, respectively. Due to the characteristic structure of the coefficient matrix, an efficient fast iterative algorithm is applied to our schemes with the computational complexity of only O( N log N ). Numerical experiments are used to verify our one-dimensional and two-dimensional theoretical analysis.