An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers' Equation

Burgers' equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers' equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers' equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications.