An improvement of convergence of a dispersion-relation preserving method for the classical Boussinesq equation

作者：Jang T S^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2018, 56: 144-160.

DOI：10.1016/j.cnsns.2017.07.024

• Dispersion-relation preserving method
• The classical Boussinesq equation
• Region of convergence
• High nonlinear dispersive wave

A dispersion-relation preserving (DRP) method, as a semi-analytic iterative procedure, has been proposed by Jang (2017) for integrating the classical Boussinesq equation. It has been shown to be a powerful numerical procedure for simulating a nonlinear dispersive wave system because it preserves the dispersion-relation, however, there still exists a potential flaw, e.g., a restriction on nonlinear wave amplitude and a small region of convergence (ROC) and so on. To remedy the flaw, a new DRP method is proposed in this paper, aimed at improving convergence performance. The improved method is proved to have convergence properties and dispersion-relation preserving nature for small waves; of course, unique existence of the solutions is also proved. In addition, by a numerical experiment, the method is confirmed to be good at observing nonlinear wave phenomena such as moving solitary waves and their binary collision with different wave amplitudes. Especially, it presents a ROC (much) wider than that of the previous method by Jang (2017). Moreover, it gives the numerical simulation of a high (or large-amplitude) nonlinear dispersive wave. In fact, it is demonstrated to simulate a large-amplitude solitary wave and the collision of two solitary waves with large-amplitudes that we have failed to simulate with the previous method. Conclusively, it is worth noting that better convergence results are achieved compared to Jang (2017); i.e.,they represent a major improvement in practice over the previous method.

• 出版日期2018-3