Convergence of a Finite Difference Scheme for the Camassa-Holm Equation

@article{Holden2006ConvergenceOA,
  title={Convergence of a Finite Difference Scheme for the Camassa-Holm Equation},
  author={H. Holden and X. Raynaud},
  journal={SIAM J. Numer. Anal.},
  year={2006},
  volume={44},
  pages={1655-1680}
}
We prove that a certain finite difference scheme converges to the weak solution of the Cauchy problem on a finite interval with periodic boundary conditions for the Camassa–Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u|_{t=0}=u_0\in H^1([0,1])$. Here it is assumed that $u_0-u_0''\ge0$, and in this case the solution is unique, globally defined, and energy preserving. 
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