A numerical study of variational discretizations of the Camassa–Holm equation

We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa–Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.