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.

This paper investigates the numerical solution for Allen-Cahn equations with perturbation parameters and strong nonlinear terms in general computational domains. To get a global second-order accuracy, we use the second-order semi-implicit Runge-Kutta (SIRK) method and some implicit-explicit (IMEX) methods for time discretization. Then the original problem is transformed into boundary value problems (BVPs) of a modified Helmholtz equation at each time step, which can be solved by a Cartesian grid-based kernel-free boundary integral (KFBI) method. In the KFBI method, the BVPs are reformulated into a corresponding boundary integral equation and then solved iteratively by a class of subspace methods such as the matrix-free generalized minimal residual(GMRES) method, while integrals involved are regarded as solutions to their equivalent interface problems. Unlike traditional boundary integral methods, this method avoids numerical integration of the singular or nearly singular integrals. Instead, it utilizes grid-based operations as an alternative to direct evaluation. Therefore, integral evaluation only requires solving equivalent but much simpler interface problems in a bounding box so that fast elliptic solvers such as fast fourier transforms(FFTs) and geometric multigrid methods are applicable. This makes the KFBI method accurate and efficient when solving constant coefficient elliptic problems in general irregular domains. It can be seen that the accuracy of the present method is verified by numerical examples.