Efficient spectral-Galerkin method for the two-dimensional Gray-Scott model

The Gray-Scott (GS) model is a nonlinear reaction diffusion system widely used in applied sciences. This paper delves into the investigation of a spectral method for the GS model with homogeneous Neumann boundary conditions. The proposed numerical scheme integrates a spectral approach based on generalized Jacobi polynomials for spatial discretization with the two-step backward differentiation formula (BDF2) for temporal discretization. We prove the boundedness, generalized stability, and convergence of this new method. Extensive numerical results demonstrate the efficiency of the new proposed scheme and provide numerical validation of the theoretical analysis. The key advantages of our new approach are twofold: (i) it utilizes generalized Jacobi polynomials with indices α = β = −2, which naturally satisfy the boundary conditions and thereby not only simplify the theoretical analysis but also yield a well-conditioned discrete system that enhances computational efficiency; and (ii) the numerical errors exhibit exponential decay in space.