The Difference Method And Characteristic Finite Element Method For Nonlinear Evolution Equation

In this paper,we mainly discuss the numerical method and the nonlinearstability of the nonlinear evolution equation called KDV equation.the di?erencemethod and the characteristics finite-element method for the KDV equation areproposed. Their nonlinear stability also are discussed in this paper.Then , thetheoretical analysis and the numerical experiments were introduced. In thisarticle,we adopt the method which is introduced in the book written by GuoBenyu to verify the nonlinear stability of the di?erence scheme of the KDVequation. Then we analyze the characteristics finite-element method, its optimalerror estimates in H1 norm is also derived. The theoretical analysis and numericalexperiments show that the two method are e?ective in solving the model.The work consists five chapters,the basic model:where,εis the di?usion coe?cient.In chapter 1,we mainly discuss the condition of the studying of the KDVequation in home and abroad.In chapter 2,we consider the di?erence scheme to compute the model problemwhere, In 2.2,we introduce the proving of the di?erence scheme and the error estimateof the KDV equation.Last, we obtain two theorems :Theorem 2.1 If u is su?cient smoothed,the order of the di?erence scheme(2.2) of the KDV equation is second order.Theorem 2.2 The di?erence scheme (2.2) of the KDV equation is stability.In chapter 3,we present the characteristics finite-element scheme, when itsatisfy the coe?cient hypothesis (3.2): then,we analyze the stability and the error estimate of the scheme,and we obtaina theorems:Theorem 3.3 We hypothesize u and uh are the solution of (3.1),(3.5) ,andsatisfy the coe?cient hypothesis (3.2),thenIn chapter 4,we give a numerical experiment ,and analyze the solution of theexperiment.In chapter 5,we make a work summary of this papaer.