International Journal of Nonlinear Analysis and Applications
Numerical solutions the nonlinear Burgers-Fisher and Burgers' equations with adaptive numerical method
Document Type : Research Paper
Author
Department of Applied Mathematics, Payame Noor University (PNU), P.O.Box 19395-4697, Tehran, Iran
Abstract
In this paper, a numerical method for finding the numerical solution of the Burgers-Fisher and Burgers' nonlinear equations is proposed. These equations are very important in many physical problems such as fluid dynamics, turbulence, sound waves and etc. We describe a meshless method to solve the nonlinear Burgers’ equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method. In this method, the moving least squares (MLS) method is used for the spatial part and the exponential time differencing(ETD) is used for the time part. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives, and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, Approaches have been stated. Method MLS provided good results for these equations due to its high flexibility and high accuracy and having a moving window, and obtains the solution at the shock point without any false oscillations. The method is described in detail, and a number of computational examples are presented. The accuracy of the proposed method is demonstrated by several test simulations.
Keywords
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Volume 14, Issue 10
October 2023Pages 257-282
- Receive Date: 14 December 2022
- Accept Date: 23 January 2023