Optimized Computational Solver for the Quasi-Linear Burgers Equation Using Legendre Pseudospectral Technique

• Published in: International journal of applied and computational mathematics Vol. 11; no. 3
• Main Authors: Singh, Harvindra, Balyan, Lokendra
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.06.2025; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Applications of Mathematics; Burgers equation; Collocation methods; Computational efficiency; Computational Science and Engineering; Computing costs; Derivatives; Fluid dynamics; High Reynolds number; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nonlinear differential equations; Nonlinear dynamics; Nuclear Energy; Operations Research/Decision Theory; Ordinary differential equations; Original Paper; Partial differential equations; Polynomials; Runge-Kutta method; Theoretical; Runge–Kutta method; 35G31; 65M70; 35C11; Error norms; Spectral collocation method; Legendre polynomials; Viscous Burgers equation; 35C07
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-025-01912-y

The Burgers equation, a renowned model in the study of fluid dynamics and nonlinear partial differential equations, continues to pose significant challenges, particularly due to its complex behavior at high Reynolds numbers (i.e., low viscosity). This paper introduces an advanced high-order spectral collocation method based on Legendre polynomials, featuring an efficient algorithm for derivative computation. The technique employs Legendre-Gauss-Lobatto (LGL) points to construct Legendre differentiation matrices, which exhibit specific symmetry properties. By leveraging these symmetries, the derivative approximation is achieved by decomposing the solution vector into its even and odd components. This novel approach effectively reduces the computational cost of derivative evaluation by nearly half. The discretization of the governing equation results in a system of ordinary differential equations (ODEs), which is solved iteratively using the fourth-order Runge–Kutta method, ensuring both stability and accuracy. A comprehensive comparison with existing models and methods is conducted to evaluate the effectiveness of the proposed approach. The results demonstrate that the technique significantly improves accuracy and computational efficiency.