Convergence of a finite difference scheme for the Camassa-Holm equation

摘要

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) - uu(xxx) = 0 with initial data u vertical bar(t)= 0 = u(0) is an element of H-1([0, 1]). Here it is assumed that u(0) - u(0)'' >= 0, and in this case the solution is unique, globally defined, and energy preserving.