A new algorithm based on Jacobian free Newton - Krylov method and Bessel functions for viscous Burger equation

Nikarya, Mehran

doi:10.22034/ijiems.2024.481917.1067

A new algorithm based on Jacobian free Newton - Krylov method and Bessel functions for viscous Burger equation

Document Type : Original Article

Author

Mehran Nikarya

Department of Electrical Engineering and Information Technology, Iranian Research Organization for Science and Technology: Tehran, Iran

10.22034/ijiems.2024.481917.1067

Abstract

The Burgers' equation is a fundamental nonlinear partial differential equation (PDE) that describes various phenomena in fluid dynamics, nonlinear acoustics, Optics and Photonics, Wave Propagation, and traffic flow. Actually viscous Burger equation is a simplified version of famous Navier-Stokes equation for incompressible fluid, without pressure gradient and external forces. In this paper a new method will be introduced to solve this nonlinear partial differential equation based on Bessel functions and the Jacobian free Newton method combined with generalized minimum residual (JFNGMRes) method. In this work the nonlinear Burger equation is converted to a nonlinear system of algebraic equations without any linearization and descretization methods. Finally, by using JFNGMRes, the solution of this nonlinear system will be achieved. To show the reliability and applicability of the proposed method, we solve some examples of the 1D and the 2D nonlinear Burger equation and then we compare our results with some famous methods in other papers and exact solutions.

Keywords

Nonlinear viscous Burger equation

Bessel function of the first kind

Spectral collocation method

Jacobian free Newton method