Two implicit finite difference schemes for solving the two-dimensional multi-term time-fractional diffusion equation with variable coefficients are considered in this paper. The orders of the Riemann–Liouville fractional time derivatives acting on the spatial derivatives can be different in various spatial directions. By integrating the original partial differential equation with time variable first, and the second-order spatial derivatives are approximated by the central difference quotients, then the fully discrete finite difference scheme can be obtained after the right rectangular quadrature formulae are used to approximate the resulting time integrals. The convergence analysis is given by the energy method, showing that the difference scheme is first-order accurate in time and second order in space. Based on a second-order approximation of the Riemann–Liouville fractional derivatives using the weighted and shifted Grünwald difference operator, we present the Crank–Nicolson scheme and prove it is second-order accurate both in time and space. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. Numerical schemes and theoretical analysis can be generalized for the three-dimensional problems.

本文考虑了求解变系数二维多项时间分数扩散方程的两种隐式有限差分格式。作用于空间导数的黎曼-刘维尔分数时间导数的阶数在不同的空间方向上可能不同。先将原偏微分方程与时间变量积分，用中心差商逼近二阶空间导数，再用直角求积公式逼近所得时间，得到完全离散的有限差分格式积分。能量法给出了收敛性分析，表明差分格式在时间上一阶准确，在空间上二阶准确。基于使用加权和移位 Grünwald 差分算子的 Riemann-Liouville 分数阶导数的二阶近似，我们提出了 Crank-Nicolson 方案并证明它在时间和空间上都是二阶准确的。提供了数值结果来验证两种所提出算法的准确性和效率。数值方案和理论分析可以推广到三维问题。