A conservative discontinuous Galerkin method for the Degasperis-Procesi equation (Q2017222)

This paper is concerned with the following two equations: The Degasperis-Procesi (DP) equation \[ u_t-u_{xxt}=uu_{xxx}+3u_xu_{xx}-4uu_x, \] and the Camassa Holm (CH) shallow water equation \[ u_t-u_{xxt}=uu_{xxx}+2u_xu_{xx}-3uu_x. \] The CH equation has three invariants and the DP equation has more than three invariants. Denote the second invariant in each case \(E_2\). The method developed in this paper is called DDG, with \(E_2\) being conservative or dissipative. The DDG is a kind of finite element method with spaces of polynomials of degree \(k\) in each cell, allowing an inner product. A time-discretization of the DDG scheme is performed using the Rung-Kutta method. Several peakon profiles are shown in diagrams to make a comparison with numerical values.