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

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 $u_t - \,u_{xxt} + \,3uu_{x\,} \, - 2u_x u_{xx\,} = \,0$ with initial data $u/_{t = 0} \, = u_0 \, \in \,H^1 \,([0,1]).$ Here it is assumed that $u_0 - u_0^ \, \ge \,0,$ and in this case the solution is unique, globally defined, and energy preserving.