Crank–Nicolson alternative direction implicit method for two-dimensional variable-order space-fractional diffusion equations with nonseparable coefficients

Abstract

In this paper, we propose alternative direction implicit (ADI) schemes to address the initial boundary value problem for two-dimensional variable-order fractional diffusion equations (VOFDEs). The Crank–Nicolson (CN) method and various ADI schemes employing different finite difference methods are utilized to approximate the temporal derivative and the spatial variable-order (VO) fractional derivatives, respectively, resulting in CN-ADI schemes. We present and prove theoretical results concerning the stability and convergence of these ADI schemes. Since the order of the VO derivatives depends on spatial and temporal variables, the resulting coefficient matrices from the discretization of VOFDEs are dense and lack a Toeplitz-like structure. We propose banded preconditioners to accelerate PGMRES methods for solving the resulting discretized linear systems. Numerical results demonstrate the high efficiency of the proposed ADI schemes.