An extrapolated hybrid framework for efficient simulation of coupled reaction-diffusion systems: Stability, convergence, and applications

We propose the Extrapolated Hybrid Method (EHM), a novel time-integration framework for stiff reaction-diffusion systems. The method combines Rosenbrock-type predictors, extrapolation, and a Crank-Nicolson corrector within a flexible operator-splitting structure that accommodates diverse time integrators for prediction and correction stages, balancing computational efficiency and accuracy. Two variants are developed: EHM-1, which is first-order accurate in time, and EHM-2, which achieves second-order temporal accuracy. Both methods are analyzed for consistency, stability, and convergence. EHM demonstrates enhanced stability over the conventional IMEX Euler scheme, particularly in stiff regimes. Numerical validation on problems with analytical solutions, the Brusselator, coupled Allen-Cahn dynamics, and Turing-pattern-forming systems such as the Schnakenberg, FitzHugh-Nagumo, and Gray-Scott models supports the theoretical analysis. The framework’s adaptable architecture demonstrates robustness under strong nonlinearities and capability to resolve intricate spatiotemporal structures, including spots, stripes, and bubble coalescence phenomena.