On error estimates for Galerkin finite element methods for the Camassa-Holm equation
Provides rigorous error analysis for finite element methods applied to a nonlinear dispersive wave equation, benefiting researchers in numerical analysis and wave modeling.
The paper proves optimal-order L2-error estimates for Galerkin finite element semidiscretizations of the Camassa-Holm equation and constructs a fully discrete scheme using Runge-Kutta time-stepping, validated by numerical experiments on smooth and peakon-type solutions.
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.