Unveiling Numerical Solutions of Zeldovich Model Using Collocation Method via Fourth‐Order Uniform Hyperbolic Polynomial B‐Spline

Abstract

ABSTRACT This study presents a numerical approach to the Zeldovich model using a fourth‐order uniform hyperbolic polynomial B‐spline collocation method. The Zeldovich model, relevant in combustion theory, describes flame propagation, thermal explosions, and detonation phenomena. In the proposed scheme, the time derivative is discretized with a finite difference method, spatial derivatives are approximated using the Crank–Nicolson method, and the nonlinear terms are linearized via the Rubin–Graves technique. The resulting system of algebraic equations satisfies the prescribed boundary conditions and is solved to obtain approximate solutions. Stability is established through von Neumann analysis, while accuracy and convergence are evaluated against exact solutions using error norms and convergence rates. The results demonstrate that the method captures the nonlinear dynamics of the Zeldovich equation with high accuracy and stability, providing a streamlined and efficient alternative for its numerical treatment.