A Finite Difference Scheme For Camassa-Holm Equation

In science and technology, a lot of mathematical models can be represented by partial differential equations. But the analytical form of the solition of the majority partial differential equations are difficult even impossible to obtained, therefore, the approximate methods are needed. Finite difference method is one of the common approximation methods to solve definite solution problem of the partial differential equation. Its prime step is to construct a reasonable difference scheme, and makes the approximate solution of the difference scheme preserve some certain main properties of original problem.Unfortunately, a difference scheme of high approximation precision is not bound to acquire a good approximation solution, because a reasonable difference scheme must also preserve the physical properties of the primary problem. So people usually construct a conservative difference scheme on the basis of physical laws.In this paper, the difference scheme of initial boundary value problem of Camassa-Holm equation is studied according to the physical conservation laws of this equation, two two-level difference schemes and a three-level difference scheme are given, the conservation of energy and the existence of solution of the difference equation is proved, a solution of the scheme is estimated, and its stability and convergence are proved with discrete energy analysis. At last, the iteration algorithm of the three difference schemes are given.