Nonlinear analysis and hybrid numerical simulation of a coupled Zeldovich model with Arrhenius kinetics

This paper investigates the analytical and numerical treatment of a class of nonlinear reaction–diffusion equations arising from the Zeldovich model with Arrhenius-type kinetics, relevant to combustion theory and plasma physics. We establish the existence, uniqueness, and linear stability of the steady-state solution under homogeneous Neumann boundary conditions. These analytical results provide a solid theoretical foundation for the model’s well-posedness and long-term behavior. On the computational side, we develop a hybrid finite difference–spline collocation (FD-SC) method for accurately solving the system in one and two spatial dimensions. The proposed FD–SC scheme combines the robustness and simplicity of finite difference methods with the high local accuracy of spline collocation, allowing for accurate and efficient handling of nonlinear effects and steep spatial variations. A detailed comparison with the standard finite difference method (FDM) highlights the superior accuracy and convergence of the FD–SC approach. We validate the proposed method through numerical experiments against an exact traveling wave solution and demonstrate convergence rates in both space and time using log–log plots and error norm tables. In two dimensions, we simulate and visualize the evolution of flame fronts over time, capturing key dynamical features of combustion waves. Collectively, the theoretical and numerical results establish the FD–SC method as a precise and efficient framework for studying stiff, nonlinear reaction–diffusion systems governed by Arrhenius kinetics.