Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawaha

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Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawahara equation R. Chousurin1,2 · T. Mouktonglang1,2 · B. Wongsaijai1,2 · K. Poochinapan1,2 Received: 5 September 2018 / Accepted: 3 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract The main contribution of this article is to introduce new compact fourth-order, standard fourth-order, and standard second-order finite difference schemes for solving the Kawahara equation, the fifth-order partial derivative equation. The conservation of mass only of the numerical solution obtained by the compact fourth-order finite difference scheme is proven. However, the standard fourth-order and standard second-order finite difference schemes can preserve both mass and energy. The stability is also proven by von Neumann analysis. According to analysis for numerical experiments, the order of accuracy for each scheme and the computational efficiency of the compact scheme are presented. To validate the potential of the presented methods, we also consider long-time behavior. Finally, results obtained from the compact scheme are superior than those from the non-compact schemes. Keywords Kawahara equation · Compact finite difference scheme · Solitary waves · Conserved quantities

 K. Poochinapan

[email protected] R. Chousurin [email protected] T. Mouktonglang [email protected] B. Wongsaijai [email protected] 1

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2

Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand

Numerical Algorithms

1 Introduction In the past, Kakutani and Ono [1] investigated the type of a mathematical model for an analysis of magnet-acoustic waves in cold collision–free plasma. Consequently, Hasimoto [2, 3] derived a higher order Korteweg–de Vries equation with an additional fifth-order derivative term as the Kawahara equation. The equation was constructed to study capillary–gravity waves in an infinitely long canal over a flat bottom in long waves when the Bond number is almost one third. Moreover, many physical phenomena, such as the blowing of the wind over an ocean surface and the propagation of a shallow water wave were also studied by using the Kawahara equation. Then, Kawahara [4] numerically investigated and noticed that this type of the equation generates both oscillatory and monotone solitary wave solutions. Using adjacent grid points, a derivative is approximated by the numerical techniques, such as finite difference, finite volume, and finite element methods, and the methods are much more effective for controlling complicated boundary conditions and geometries. With grid refinement, their sluggish convergence to the exact solution which requires many more grid points to execute a targeted accuracy level is the major difficulty of these methods. Regarding the mentioned problem, then researchers have developed compact higher order finite d