An Unconditionally Stable Numerical Algorithm for Two-Dimensional Convection-Diffusion-Reaction Equations - Nwaigwe, Chinedu (Journal of Applied Nonlinear Dynamics, 2026)
An unconditionally stable finite difference scheme is developed and fully analyzed for two-dimensional convection-diffusion-reaction equations with nonlinear coefficients and external sources. The well-known difficulty in obtaining a stable second-order discretization of the convection term is resolved by first adopting a central discretization in space. The semi-discrete convection term is split into the positive and negative parts. Then a non-standard time-integration is proposed - by treating one part implicitly and the other explicitly. This allowed to maintain numerical stability, retain compactness of the stencil and avoid division by the diffusion coefficient, hence can be applied to purely convection (zero diffusion) problems. The solvability, consistency and stability of the scheme are established. Numerical results are provided to verify the theoretical properties.