A strong second-order two-stage explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for two-dimensional Gray-Scott model

TL;DR

This paper introduces a second-order explicit/implicit spectral Galerkin finite element method for the 2D Gray-Scott model, achieving high accuracy, stability, and efficiency through a novel combination of techniques.

Contribution

It develops a unconditionally stable, high-order accurate two-stage method combining spectral orthogonal basis and Galerkin finite elements for the Gray-Scott model.

Findings

Method is unconditionally stable and second-order accurate in time.

Spectral orthogonal basis minimizes spatial errors.

Numerical examples confirm theoretical stability and efficiency.

Abstract

This paper proposes a strong second-order two-step explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for solving a two-dimensional Gray-Scott model subject to appropriate initial and boundary conditions. The constructed approach discretizes at the first stage utilizing a second-order explicit method while a second-order implicit scheme is employed at the second phase. The space derivatives are approximated with the Galerkin finite element formulation combined with a spectral orthogonal basis. With this combination, the errors increased at the first stage are balanced by the errors decreased at the second phase so that the stability is maintained. Furthermore, the use of the spectral orthogonal basis minimizes the space errors. Thus, the new computational approach calculates efficiently numerical solutions and preserves a strong stability and high-order accuracy. The theoretical studies indicate that the proposed strategy is unconditionally stable, temporal second-order accurate and spatial $qth$-order convergent using the $L^{\infty}(0,T;[L^{\infty}(\Omega)]^{2})$-norm, where $q$ is an integer greater than or equal $2$. Some numerical examples are performed to confirm the theory and to demonstrate the efficiency of the developed algorithm.