2018/05/28 by Antonopoulos, D. C., Dougalis, V. A., Mitsotakis, D. E. · 1 citation
#35Q53 #65M60 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1805.10744
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 L2-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.