Year: 2018
Type: preprint
Abstract: We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type.
Source: arXiv (Cornell University)
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funders: Royal Society Te Apārangi, Marsden Fund
Awards: G1193341635, G2052991896
OpenAlex ID: W4292536828