Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications

In this study, a class of nonlinear heterogeneous reaction–diffusion system (RDS) has been considered that arises in modeling epidemiological interactions, environmental sciences, and chemical and ecological systems. Numerical and analytic solutions for this kind of variable medium nonlinear RDS are challenging. This article developed a few highly accurate numerical schemes for such problems. For the spatial integration of the heterogeneous RDS, a few finite difference schemes, a Bernstein collocation scheme, and a Fourier transform scheme were employed. The stability and accuracy analysis of the spectral schemes were studied to confirm the order of convergence of the approximation. A few methods of lines were then used for the temporal integration of the resulting semidiscrete model. It was confirmed theoretically that the spectral/pseudo-spectral method is very efficient and highly accurate for such a model. A fast and efficient solver for the resulting full discrete system is highly desired. A Newton–GMRES–Multigrid solver was applied for the resulting full discrete system. It is demonstrated in tabular form that a multigrid accelerated Newton–GMRES solver outperforms most linear solvers for such a model.