A conservative local discontinuous Galerkin method with spectral accuracy for the Degasperis-Procesi equation

This paper introduces a novel conservative local discontinuous Galerkin scheme, combined with spectral element discretization, aiming to tackle the Degasperis-Procesi (DP) equation in higher-order accuracy. Tailored numerical fluxes are formulated to ensure the preservation of mass and energy in the sense of semi-discretized format. By employing the Crank-Nicolson time integration, the conservation laws for the fully-discrete scheme are rigorously demonstrated. Numerical simulations encompassing accuracy test, shock capture, invariant preservation and comparisons with the conventional total variation diminishing (TVD) Runge-Kutta time discretization are conducted for peakon and shockpeakon solutions. The simulations comprehensively assess the proposed approach, revealing its performance advantages.