Efficient and accurate temporal second-order numerical methods for multidimensional multi-term integrodifferential equations with the Abel kernels

Abstract

This work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three-dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high-dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order ${\tau }^2+h_1^4+h_2^4$, where $\tau$, $h_1$, and $h_2$ denote the time-space step sizes, respectively, and the second scheme converges to the space-time second-order accuracy. Finally, the numerical results verify the correctness of the theoretical analysis and show that the method of this article is competitive with the existing research work.