Energy-preserving $H^1$-Galerkin schemes for the Hunter--Saxton equation

We consider the numerical integration of the Hunter--Saxton equation, which models the propagation of weakly nonlinear orientation waves. For the equation, we present two weak forms and their Galerkin discretizations. The Galerkin schemes preserve the Hamiltonian of the equation and can be implemented with cheap $H^1$ elements. Numerical experiments confirm the effectiveness of the schemes.