Energy-preserving H1H1-Galerkin schemes for shallow water wave equations with peakon solutions

New energy-preserving Galerkin schemes for the Camassa–Holm and the Degasperis–Procesi equations which model shallow water waves are presented. The schemes can be implemented only with cheap H1H1 elements, which is expected to be sufficient to catch the characteristic peakon solutions. The keys of the derivation are the Hamiltonian structures of the equations and an L2L2-projection technique newly employed in the present Letter to mimic the Hamiltonian structures in a discrete setting, so that the desired energy-preserving property rightly follows. Numerical examples confirm the effectiveness of the schemes.

► Numerical integration of the Camassa–Holm and Degasperis–Procesi equation.
► New energy-preserving Galerkin schemes for these equations are proposed.
► They can be implemented only with P1 elements.
► They well capture the characteristic peakon solutions over long time.
► The keys are the Hamiltonian structures and L2L2-projection technique.