Finite Difference Method Of A Nonlinear Equation

Finite difference method, also named after grid 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 we construct a difference scheme on the basis of physical laws, and name it conservative scheme.In this paper, we constructed the new conservative schemes of regularized long wave equation according to the physical laws of this equation. In Chapter 1 we mainly expatiated on the finite difference method and conservative scheme, and reviewed some researches of regularized long wave equation. Chapter 2,3 and 4 are the main bodies of this paper. In Chapter 2, a new conservative three-levels finite difference scheme for initial value and boundary value problem of regularized long wave equation was considered, whose truncation error was O(Ï„~2 + h~2). And the convergence and stability were proved. Numerical examples showed that the methods had good accuracy and efficiency. In Chapter 3,a new conservative two-levels implicit finite difference scheme for initial value and boundary value problem of generalized regularized long wave equation was considered, whose truncation error was O (Ï„~2 + h~2). And the convergence and stability were proved. Numerical examples showed that the methods had good accuracy and efficiency. In Chapter 4, a new conservative three-levels finite difference scheme for initial value and boundary value problem of generalized regularized long wave equation was considered, whose truncation error was O (Ï„~2 + h~2). And the convergence and stability were proved. Numerical example shows that our scheme has higher accuracy and good practicability. In Chapter 5 we briefly reviewed the whole paper and put forward the further tasks.