In this paper, an alternating direction implicit (ADI) compact difference scheme will be proposed for solving semilinear time-fractional mobile–immobile equations in two dimensions. In the time direction, the first derivative and Caputo derivative term are treated by means of second-order backward differentiation formula (BDF) and L1 discrete method, respectively. Then, the nonlinear term is linearized and a fully discrete ADI compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are established by the discrete energy method. In addition, the accuracy and effectiveness of the scheme are illustrated by several numerical experiments.

在本文中，将提出一种交替方向隐式（ADI）紧致差分格式，用于求解二维半线性时间分数阶移动-不动方程。在时间方向上，分别通过二阶后向微分公式（BDF）和L 1离散方法处理一阶导数和Caputo导数项。然后，将非线性项线性化，并通过四阶紧致逼近与空间离散化构建完全离散的ADI紧致差分方案。稳定性和收敛性是通过离散能量方法建立的。另外，通过几个数值实验说明了该方案的准确性和有效性。