Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

Abstract

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

Keywords:

• Classical long wave system
• high-order scheme
• local discontinuous Galerkin method
• classical long wave system
• strong-stability-preserving Runge Kutta method

Mathematics Subject Classification 2010:

• 35G50
• 47J35
• 65N30 PACS: 02.60.-x
• 02.30.Jr

Acknowledgements

The authors would like to thank the learned anonymous reviewers for their insightful feedback and recommendations regarding enhancing the quality of our manuscript. The first author would also like to acknowledge his supervisor, Dr. Jugal Mohapatra, NIT Rourkela, for assisting in the revision of the manuscript and composing the responses.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

This manuscript incorporates all the data that were generated or analyzed during this study.

Additional information

Funding

This research is supported by the “University Grants Commission (UGC),” (NTA Ref. No. 191620043696).