Advanced Numerical Treatment of Newell Whitehead Segel Equation Using the Method of Lines with Third-Order Finite Difference Approximations

Abstract

The Newell-Whitehead-Segel equation (NWSE) is a foundational nonlinear model for understanding pattern development and bifurcation in a variety of physical and engineering systems, such as Rayleigh-Bénard convection, material microstructure evolution, and nanostructure selfassembly. This study proposes a strong high-order numerical technique for solving the NWSE that combines the Method of Lines with thirdorder finite difference approximations for spatial derivatives. The spatial discretization transforms the governing partial differential equation into a system of ordinary differential equations, which are then integrated in time using the standard fourth-order Runge-Kutta technique. A thorough stability and convergence analysis is carried out to determine the theoretical validity of the proposed method. Numerous numerical studies show that the approach is highly accurate, stable, and computationally advantageous across a number of examples of testing. This work makes a novel contribution by constructing third-order one-sided finite-difference stencils at the boundaries, which ensure high-order accuracy while successfully implementing Dirichlet boundary conditions and avoiding precision loss near domain boundaries. The suggested numerical framework is a reliable and effective tool for describing challenging pattern-forming systems, as well as precisely parametric studies for design and control applications in engineering and scientific studies.OPEN ACCESS Received: 01/01/2026 Accepted: 27/02/2026

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