Abstract
In this work, we design, analyze, and numerically test an invariant preserving discontinuous Galerkin method for solving the nonlinear Camassa--Holm equation. This model is integrable and admits peakon solitons. The proposed numerical method is high order accurate, and preserves two invariants, momentum and energy, of this nonlinear equation. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The numerical simulation results for different types of solutions of the Camassa--Holm equation are provided to illustrate the accuracy and capability of the proposed method.
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