In this paper, we propose a second-order implicit–explicit (IMEX) time-stepping scheme for the Gray-Scott (GS) model. In order to achieve a fast, stable and efficient scheme in time, we develop a linear second-order time-stepping scheme for solving GS model. We prove the linear stability and convergence of the proposed scheme. The advantage of our numerical scheme is stable, that is, there is no need for other stability restrictions to the nonlinear terms, only a relatively small time step is needed. Moreover, we prove that the numerical solutions achieve second-order accuracy in the time. Finally, several numerical experiments are conducted to verify the accuracy both in space and time of the numerical scheme. Numerical examples illustrate the accuracy and efficiency of the IMEX scheme is effective for the GS model.

在本文中，我们为 Gray-Scott (GS) 模型提出了一种二阶隐式-显式 (IMEX) 时间步长方案。为了及时实现快速、稳定和高效的方案，我们开发了一种线性二阶时间步长方案来求解 GS 模型。我们证明了所提出方案的线性稳定性和收敛性。我们的数值方案的优点是稳定，即不需要对非线性项进行其他稳定性限制，只需要相对较小的时间步长。此外，我们证明了数值解在时间上达到了二阶精度。最后，进行了多次数值实验，以验证数值方案在空间和时间上的准确性。数值例子说明IMEX方案的准确性和效率对GS模型是有效的。