Published by De Gruyter June 8, 2016

# Qualitative analysis and numerical solution of burgers’ equation via B-spline collocation with implicit euler method on piecewise uniform mesh

• Vikas Gupta and Mohan K. Kadalbajoo

## Abstract

In the present paper, a numerical method is proposed for solving one-dimensional time dependent Burgers’ equation for various values of Reynolds number on a rectangular domain in the x-t plane. For large values of Reynolds number, a boundary layer produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a quasi-linear singularly perturbed parabolic problem. Classical numerical methods with uniform mesh are known to be inadequate to solve such problems, unless an unacceptably large number of mesh points are used. Therefore, in order to overcome this drawback associated to classical numerical methods with uniform mesh, we construct a numerical scheme that comprises of implicit Euler method to discretize in temporal direction on uniform mesh and a B-spline collocation approach to discretize the spatial variable with piecewise uniform Shishkin mesh. Quasi-linearization process is used to tackle the non-linearity and shown that quasi-linearization process converges quadratically. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. These bounds are applied in convergence analysis of the proposed method on Shishkin mesh. The method has been shown to be first order convergent in the temporal variable and almost second order accurate in the spatial variable. Higher accuracy and convergence of the method is demonstrated by numerical examples and an estimate of the error is given. Comparisons of numerical results are made with others available in the literature for both low as well as high Reynolds numbers.

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