A fast temporal second-order compact finite difference method for two-dimensional parabolic integro-differential equations with weakly singular kernel

We propose a fast temporal second-order compact finite difference method for solving a class of two-dimensional parabolic integro-differential equations with weakly singular kernel. The sum-of-exponentials approximation of the kernel function is combined with the product averaged integration rule to approximate the weakly singular integral term, and the fourth-order compact finite difference method is employed to discretize the spatial derivative operator. We use the discrete energy technique to rigorously prove that the proposed fast method is almost unconditionally stable and convergent. Compared with the direct method, the fast method significantly reduces storage and computational costs while maintaining the temporal second-order convergence and spatial fourth-order convergence for weakly singular solutions. Some comparisons with the fast method using the exponential-sum-approximation technique, the fast Runge–Kutta convolution quadrature method and the explicit fast method based on the sum-of-exponentials approximation are discussed. The numerical results confirm the results of theoretical analysis and demonstrate the computational efficiency of the fast method.