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

Date

2019-10-03

Author

Elgindy, Kareem T.
Karasözen, Bülent

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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 B --> 0.

Subject Keywords

Barycentric, Burgers' equation, Discontinuous Galerkin, Gegenbauer-Gauss, Gegenbauer polynomials, Integration matrix

URI

https://hdl.handle.net/11511/30352

Journal

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS

DOI

https://doi.org/10.1080/00207160.2018.1554860

Collections

Graduate School of Applied Mathematics, Article

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Citation Formats

K. T. Elgindy and B. Karasözen, “High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers’ equation,” INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, pp. 2039–2078, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30352.