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

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