Robust numerical scheme for nonlinear modified Burgers equation

ABSTRACT

A new and efficient numerical scheme for approximating one-dimensional nonlinear modified Burgers equation has been illustrated in this study. The method makes use of collocation of quintic splines and Crank Nicolson scheme for spatial and temporal discretization, respectively. Linearization involves a quasilinearization process. Since splines are smooth functions which do not exhibit oscillations associated with polynomials and possess easily computable derivatives, working with splines has a computational edge over other existing methods. The method is unconditionally stable, which is justified by von Neumann stability analysis. Convergence analysis of the method is performed and it is proved that the method has fourth-order convergence in space and second order in time. Performance of the method is assessed by computing the CPU time, and small values of CPU time ascertains that the method is highly time efficient. Proposed method yields good numerical results for moderate values of viscosity. Efficiency of the scheme is established from the fact that the $L_2$ and $L_{\mathrm{\infty }}$ error norms are minute. Results achieved using the proposed numerical scheme are more precise and reliable as compared to many existing works on modified Burgers equation. This method is robust, fast, flexible and easy to implement.

KEYWORDS:

• Modified Burgers equation
• quintic B-spline
• collocation
• Crank Nicolson scheme
• quasilinearization

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N06
• 65N12
• 65N35
• 65D07

Acknowledgments

Authors are very grateful to the reviewers for their valuable comments and suggestions which has helped immensely in improving the quality of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Authors also acknowledge the support from National Institute of Technology, Calicut for funding this research work.