Turing patterns across geometries: A proven DSC-ETDRK4 solver from plane to sphere

This paper presents a unified and robust numerical framework that combines the Discrete Singular Convolution (DSC) method for spatial discretization with the Exponential Time Differencing Runge–Kutta (ETDRK4) scheme for temporal integration to solve reaction–diffusion systems. Specifically, we investigate the formation of Turing patterns – such as spots, stripes, and mixed structures – in classical models including the Gray–Scott, Brusselator, and Barrio–Varea–Aragón–Maini (BVAM) systems. The DSC method, employing the regularized Shannon's delta kernel, delivers spectral-like accuracy in computing spatial derivatives on both regular and curved geometries. Coupled with the fourth-order ETDRK method, this approach enables efficient and stable time integration over long simulations. Importantly, we rigorously establish the necessary theoretical results – including convergence, stability, and consistency theorems, along with their proofs – for the combined DSC-ETDRK4 method when applied to both planar and curved surfaces. We demonstrate the capability of the proposed method to accurately reproduce and analyze complex spatiotemporal patterns on a variety of surfaces, including the plane, sphere, torus, and bumpy geometries. Numerical experiments confirm the method's versatility, high accuracy, and computational efficiency, making it a powerful tool for the study of pattern formation in reaction–diffusion systems on diverse geometries.