Convergence analysis and error estimate of second-order implicit–explicit scheme for Gray-Scott model

Abstract

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.

Keywords:

• Gray-scott model
• IMEX scheme
• consistency analysis
• stability
• error estimation

2010 Mathematics Subject Classifications:

• 34A45
• 34D20
• 65Gxx

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Fangying Song was supported by the National Natural Science Foundation of China (No.11901100). The work of Jun Zhang is supported by the National Natural Science Foundation of China (Nos. 12026241, 12026240, 11901132 and 62062018), Guizhou Key Laboratory of Big Data Statistics Analysis (No. BDSA20200102) and Natural Science Research Projects of Education Department of Guizhou Province (KY [2021]015).