High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers' equation

ABSTRACT

We present a high-order integral nodal discontinuous Galerkin (DG) method to solve Burgers' equation. The method lays the first stone of a novel class of integral nodal DG methods exhibiting exponential convergence rates in both spatial and temporal directions; thus, producing highly accurate approximations using a significantly small number of collocation points. This useful result is proven theoretically under some mild conditions. The paper also introduces the first rigorous rounding-error analysis for the Gegenbauer integration matrices proving their stability feature. Two useful strategies were proposed to significantly reduce the errors in certain special cases and to handle problems with relatively large time domains. Extensive numerical comparisons with other competitive numerical methods manifest the superior accuracy and efficiency of the proposed numerical method. The established numerical method is so accurate in general for sufficiently smooth solutions to the extent that exact, or nearly exact solutions can be achieved using relatively small collocation points as the viscosity parameter $\nu \to 0$.

Keywords:

• Barycentric
• Burgers' equation
• discontinuous Galerkin
• Gegenbauer-Gauss
• Gegenbauer polynomials
• integration matrix

2010 Mathematics Subject Classifications:

• 41A10
• 65G50
• 65M12
• 65M70

Acknowledgements

We would like to thank the Editor for handling the manuscript. We would like also to thank the Reviewers for the comments and suggestions that helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Kareem T. Elgindy  http://orcid.org/0000-0001-7369-4988Bülent Karasözen  http://orcid.org/0000-0003-1037-5431

Notes

1 We added the word ‘extended’, because the used transformation is an extension to the original Cole-Hopf transformation when c=1.

Additional information

Funding

The first author would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through project No. SR161013.