Numerical Method For Some Kinds Of Partial Differential Equations About Solitary Wave

This Master thesis consists of five chapters in all.We will have a simple discussion about solitary wave—the physical back-ground of the partial differential equations referred in this thesis.Then we spread the thesis into the second chapter around the following equa-tion: which we are familiar with called linear Sobolev equation. Some kinds of finite element methods(references [1][2]) have got the theoretical results systematically about this problem,but we will apply the generalized finite difference scheme which is also named finite volume element to this question.Reference [3] has done some researches in semi-discrete about this equation. Based on the work of reference [6], we put forward two kinds of full-discrete schemes in this chapter. With the strict analysis to these schemes, we get the most optimal order error estimate in H1 norm. And the numerical example verifies the theoretical results.In the third chapter, we will talk about a kind of typical equation having some-thing with solitary wave. The mathematical model is: ut+εupux+γuxxx-δuxxt=0 which is called as generalized Improved KdV equation (in fact, when p=1,γ= 1,δ=0, the above equation is the common KdV). When the above parameters take different values,this equation will be changed into different solitary wave equations as well as with different numerical methods (references [4][5][6][7][8][9]).In this chapter, we use a linear implicit difference conservation scheme to deal with the question. Also with strict analysis to the scheme, we can prove its conservative property and unconditional stability of the numerical scheme and get the order of the error estimate in L∞norm. The numerical example shows the theoretical results. The following equation is what we will discuss in the fourth chapter: ut-uxxt+uxxxxt+ux+(up)x=0,(xl<x<xr,0<t<T) which is called as generalized Rosenau-RLW equation. It also can be changed into many different equations because of the existence of the variable parameter-s.References[10] [11][12][13] show us some deformation forms as well as the cor-responding numerical methods.In this chapter, we propose an implicit conservative difference scheme to deal with the question.With strict analysis to the scheme, we can also prove its conservative properties and unconditional stability of the numer-ical scheme and get the order of the error estimate in L∞norm as what we do in the chapter 3. The numerical example shows the theoretical results.In the last chapter, I do some prospects with the contents of chapter 2.At present,the theories about linear system have already been classical.And we also know that the linear system describes the intricate nature phenomenon so approx-imately that it can not be accurate enough.For all of that,the author hopes to add a nonlinear term to the linear Sobolev equation referred in the Chapter 2. First,we change the Sobolev equation into the following form: ut-(φ(x, t) uxt)x-αuxx+f(u)x=0 In this way,we can put forward our job based on the work of Chapter 2 with the generalized difference method.The work is on my way and we wish a satisfying result could come to us in the near future.