Numerical solutions the nonlinear Burgers-Fisher and Burgers' equations with adaptive numerical method

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Najafzadeh Neda

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Journal Paper

Paper Information

Journal: INTERNATIONAL JOURNAL OF NONLINEAR ANALYSIS AND APPLICATIONS Year:2023 | Volume:14 | Issue:10 Page(s): 257-282

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Title

Numerical solutions the nonlinear Burgers-Fisher and Burgers' equations with adaptive numerical method

Author(s)

Najafzadeh Neda | Issue Writer Certificate

Keywords

‎Adaptive Numerical Analysis

Burgers' nonlinear equation

Moving Least Squares (MLS)

Exponential Time Differencing (ETD)

Meshless Method

Burgers-Fisher equation‎

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.

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