Exploring pattern formation dynamics in the Gray–Scott model: A finite volume numerical investigation into intricacies

This research aims to present a comprehensive investigation of the self-organization of two-dimensional patterns composed of numerous dots and lines using numerical techniques, specifically focusing on the Gray–Scott (GS) model, a well-established framework for studying reaction–diffusion systems. The staggered grid finite volume approach is utilized and the Crank–Nicholson method (θ=12⋅) is used for time discretization in our simulations, as it is known to yield a reliable and accurate solution. The present study involves an investigation into the stability of the temporal semi-discrete numerical scheme, along with a comprehensive analysis of the errors through the utilization of standard solutions that demonstrate second-order convergence in both temporal and spatial domains. The utilization of pattern formulations proves to be extremely useful in the analysis of the interaction between diffusion and reactions. This is accomplished by solving the coupled partial differential system using finite difference techniques, which serve to enhance accuracy while simultaneously preserving the stability of the system. To further examine the system’s behavior, the finite volume methodology (FVM) was utilized to examine the dynamics of the GS model. The assessment of stability is performed on fixed points, wherein analytical solutions are compared to computational techniques such as the Alternate Direction Implicit methodology and the FVM. Furthermore, we employ the FVM in spatial discretization and the Crank–Nicolson scheme in temporal discretization to solve the GS model. This study represents a novel application of two distinct methodologies to analyze error co-variances and system efficiency in a particular system. The utilization of these approaches in combination is unprecedented in the existing literature and aligns with our current comprehension of the subject matter. The results demonstrate the accuracy and reliability of the proposed numerical schemes, highlighting their potential applicability to a wide range of reaction–diffusion systems. The findings of this research have significant implications for several disciplines, including material science, chemical engineering, and biophysics, where self-organization patterns are essential to the creation of innovative materials, novel devices, and biological systems. The results of our research provide vital insight into the fundamental mechanics of pattern formation. In addition, these discoveries have the potential to be utilized in the advancement and enhancement of a wide variety of technologies and applications.