On error estimates for Galerkin finite element methods for the Camassa-Holm equation
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 -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.
Cite
@article{arxiv.1805.10744,
title = {On error estimates for Galerkin finite element methods for the Camassa-Holm equation},
author = {D. C. Antonopoulos and V. A. Dougalis and D. E. Mitsotakis},
journal= {arXiv preprint arXiv:1805.10744},
year = {2019}
}