A kernel-free boundary integral method for reaction-diffusion equations

This paper was based on a kernel-free boundary integral (KFBI) method for solving the reaction-diffusion equation. The KFBI method serves as a general elliptic solvers for boundary value problems in an irregular problem domain. Unlike traditional boundary integral methods, the KFBI method avoids complicated direct integral calculations. Instead, a Cartesian grid-based five-point compact difference scheme was used to discretize the equivalent simple interface problem, whose solution is the integral involved in the corresponding boundary integral equations (BIEs). The resulting linear system was treated with a fast Fourier transform (FFT)-based elliptic solver, and the BIEs were iteratively solved by the generalized minimal residual (GMRES) method. The first step in solving the reaction-diffusion equation was to discretize the time variable with a two-stage second-order semi-implicit Runge-Kutta (SIRK) method, which transforms the problem into a spatial modified Helmholtz equation in each time step and can be solved by the KFBI method later. The proposed algorithm had second-order accuracy in both time and space even for small diffusion problems, and the computational work was roughly proportional to the number of grid nodes in the Cartesian grid due to the fast elliptic solver used. Numerical results verified the stability, efficiency, and accuracy of the method.