Linear-implicit and energy-preserving schemes for the Benjamin-type equations

Abstract

The Benjamin-type equations are typical types of non-local partial differential equations usually describing long internal waves along the interface of two vigorously different fluid layers. In this work, we propose two kinds of novel linear-implicit and energy-preserving algorithms for the Benjamin-type equations. These algorithms are based on the invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, respectively. The IEQ and SAV are originally developed to construct energy stable schemes for the class of gradient flows. Herein, we innovate such schemes to the Benjamin-type equations and, essentially, verify them to be effective to construct energy-preserving schemes for the Hamiltonian structures. Meanwhile, numerical experiments are presented to demonstrate the efficiency of these schemes eventually.

Keywords:

• Benjamin-type equations
• energy-preserving
• invariant energy quadratization approach
• scalar auxiliary variable approach

2010 Mathematics Subject Classifications:

• 65M06
• 65M60
• 65P10

Acknowledgments

The authors also thank the China-Sri Lanka Joint Center for Education and Research, Chinese Academy of Sciences for their kind help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant number 41725017, 41590864, 11971242], National Key Research and Development Program of China [grant numbers 2016YFC0600101, 2016YFB0200801, 2018YFC1504205], Major Projects of Natural Sciences of University in Jiangsu Province of China [grant number 18KJA110003].