Higher order splitting approaches in analysis of the Burgers equation

Authors

  • Murat Sari Yildiz Technical University
  • Huseyin Tunc Dept. of Mathematics, Faculty of Arts and Science, Yildiz Technical University Istanbul 34220, Turkey
  • Muaz Seydaoglu Dept. of Mathematics, Faculty of Arts and Science, Mus Alparslan University Mus 49100, Turkey

Keywords:

Burgers equation, Strang Splitting, Galerkin method, Cubic B-spline, Extrapolation

Abstract

This article proposes some higher order splitting-up techniques based on cubic B-spline Galerkin finite element method in analysing the Burgers equation model. The strong form of both conservation and diffusion parts of the time-splitted Burgers equation have been considered in building the Galerkin approach. To integrate the corresponding ODE system, the Crank-Nicolson time discretization scheme is used. The proposed schemes are shown to be unconditionally stable. Two challenging examples have been considered with changing values of the kinematic viscosity constant of the medium. Especially, the cases of shock waves of severe gradient are solved and checked with both exact solution and the literature. The qualitative and quantitative results demonstrate that our numerical approach has far higher accuracy than the rival methods.

Author Biography

Murat Sari, Yildiz Technical University

Murat Sari was born in Adana, Turkey, in 1969. He received the B.Sc. degree from the Ondokuzmayis University, in 1991, and the M.Phil./Ph.D. degrees in Mathematics from University of South Wales, UK, in 2000.

In 1993, he joined the Department of Mathematics, Pamukkale University, as a Research Assistant. In the period 2002-2012, he was with the Department of Mathematics, Denizli, where he was an Assistant Professor, became an Associate Professor in 2012. For more than three years, he is in in Yildiz Technical University. His current research interests cover

  • Computational Methods
  • Numerical Methods
  • Numerical Solutions of Differential Equations
  • Modelling of Nonlinear Behaviours
  • Economical Modelling
  • Biomechanical/Biomedical Modelling
  • Wave Propagation Modelling.

Dr. Sari has published over 40 high quality scientific papers, more than 20 conference proceedings, written/translated various chapters in some books.

He has carried out many talks in various universities such as: Bielefeld University, Germany, Petroleum-Gas University of Ploieşti, Romania, Pardubice University, Czech Republic, Université Toulouse III - Paul Sabatier, France, Czech Technical University in Prague, Czech Republic, Ningbo University, China, Pamukkale University, Bogazici University, Yildiz Technical University, Erzincan University, Dumlupinar University.

He became reviewers for many journals for instance;

  • Medical and Biological Engineering and Computing
  • Journal of Sports Science and Medicine
  • IET Intelligent Transport Systems
  • International Journal for Numerical Methods in Fluids
  • Applied Numerical Mathematics
  • Numerical Methods for Partial Differential Equations
  • Neural Computing and Applications
  • International Journal of Computer Mathematics
  • Applied Mathematical Modelling
  • Boundary Value Problems
  • Applicable Analysis
  • ANZIAM Journal
  • Expert Systems
  • Computer Physics Communications
  • Applied Mathematics and Computation
  • Computer Methods and Programs in Biomedicine
  • Computer Applications in Engineering Education
  • African Journal of Agricultural Research
  • International Journal of Nonlinear Science
  • Journal of Computational Science
  • Bulletin of the Malaysian Mathematical Sciences Society
  • Engineering Applications of Artificial Intelligence
  • Advances in Engineering Software
  • Sains Malaysiana

Dr. Sari is currently an editor/associate editor for various scientific journals.

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Published

21-02-2019

Issue

Vol. 46 No. 1 (2019): Kuwait Journal of Science

Section

Mathematics