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 ut−uxxt +3uux−2uxuxx−uuxxx = 0 with initial data u|t=0 = u0 ∈ H1([0, 1]). Here it is assumed that u0 − u′′ 0 ≥ 0 and in this case, the solution is unique, globally defined, and energy preserving.