Development of Numerical Algorithms for Solving Nonlinear Partial Differential Equations

Zahraa Ahmed Sahib^1,*, Najmeh Malek Mohammadi¹

¹Islamic Azad University Tehran Branch, Iran

Emails: aaszween@gmail.com; Mohammadi.Najmeh@gmail.com

Abstract

This study focuses on the development of efficient numerical algorithms for solving nonlinear partial differential equations (PDEs). The research integrates theoretical analysis and practical numerical experiments to address the challenges posed by nonlinear PDEs, which often lack closed-form solutions and exhibit sensitivity to initial and boundary conditions. Benchmark models such as Burgers’ Equation, the Korteweg–de Vries (KdV) Equation, and the Navier–Stokes Equations are highlighted due to their significance in physical and engineering applications. Traditional numerical methods—Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM)—are reviewed with respect to accuracy, stability, and computational efficiency. Numerical stability concepts, including Von Neumann analysis and the CFL condition, are discussed alongside sources of error and strategies for error reduction. New algorithms were proposed by enhancing traditional schemes, incorporating adaptive mesh refinement, and integrating stability techniques. Numerical experiments on benchmark problems demonstrated improved accuracy, enhanced stability in handling nonlinear terms, and acceptable computational efficiency. The findings emphasize the importance of selecting suitable numerical methods, conducting stability analysis, and applying adaptive techniques. The study recommends higher-order schemes, conservative formulations for fluid dynamics, and double precision when necessary, ensuring reliable and reproducible computational results.

Keywords: Nonlinear PDEs; Numerical Algorithm; Finite Difference; Finite Element; Stability Analysis; Computational Mathematics