Modified Crank Nicolson Type Method for Burgers Equation

AUTHOR AND
AFFILIATION

SACHIN S. WANI
Pillai College of Engineering, New Panvel, Navi Mumbai, INDIA 410206

Email-id-sachin.swani@gmail.com

KEYWORDS:

Burgers Equation, Finite Difference, Crank Nicolson, Convergence, Stability. AMS Subject Classification: 65M12

Issue Date:

October, 2016

Pages:

ISSN:

2319-8044 (Online) – 2231-346X (Print)

Source:

Vol.28 – No.5

PDF

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DOI:

http://dx.doi.org/10.22147/jusps-A/280505

ABSTRACT:

In this paper a new finite difference scheme called Modified Crank Nicolson Type (MCNT)method is proposed to solve one dimensional non linear Burgers equation. The new scheme is obtained by discretizing the nonlinear term uux explicitly, u is approximated at t=tn+1 and ux by central difference at t = tn. The stability and convergence of the scheme is analysed. The method is shown to be first order accurate in time and second order accurate inspace. The solutions of Burgers equation obtained by MCNT are compared with the exact and numerical solutionsof Burgers equation available in the literature. For comparisons of numerical solutions with existing methods three test problems are tested.The L2 norm and L norm are used to compare the errors in the solutions. The solutions of Burgers equations are plotted at different time steps for different values of constant of diffusivity k.

Copy the following to cite this Article:

SACHIN S. WANI, “Modified Crank Nicolson Type Method for Burgers Equation”, Journal of Ultra Scientist of Physical Sciences, Volume 28, Issue 5, Page Number , 2016


Copy the following to cite this URL:

SACHIN S. WANI, “Modified Crank Nicolson Type Method for Burgers Equation”, Journal of Ultra Scientist of Physical Sciences, Volume 28, Issue 5, Page Number , 2016

Available from: http://www.ultrascientist.org/paper/597/


In this paper a new finite difference scheme called Modified Crank Nicolson Type (MCNT)method is proposed to solve one dimensional non linear Burgers equation. The new scheme is obtained by discretizing the nonlinear term uux explicitly, u is approximated at t=tn+1 and ux by central difference at t = tn. The stability and convergence of the scheme is analysed. The method is shown to be first order accurate in time and second order accurate inspace. The solutions of Burgers equation obtained by MCNT are compared with the exact and numerical solutionsof Burgers equation available in the literature. For comparisons of numerical solutions with existing methods three test problems are tested.The Lnorm and L norm are used to compare the errors in the solutions. The solutions of Burgers equations are plotted at different time steps for different values of constant of diffusivity k.