Numerical Methods for Nonlinear Partial Differential Equations: A Comparative Study

Abstract

Numerical solution of nonlinear partial differential equations is central to contemporary applied mathematics, underpinning models of fluid flow, nonlinear wave propagation, reaction–diffusion, and pattern formation. This paper presents a comparative study of three principal classes of numerical method finite differences, finite elements, and spectral methods applied to the viscous Burgers' equation as a prototype nonlinear problem. Stability, convergence, and computational cost are analysed theoretically and illustrated numerically. The study confirms the complementary strengths of the methods: finite differences offer simplicity and suitability for structured grids; finite elements accommodate complex geometries with provable convergence theory; spectral methods provide exponential convergence for smooth solutions on simple domains. The paper concludes with guidance on method selection for contemporary scientific computing.