Convergence Analysis Of Difference Schemes For Two Nonlinear Partial Differential Equations

This paper discusses the numerical solutions of the Generalized Symmetric Regular-ized Long Wave equations and Rosenau-kdv equation by finite difference methods. We construct difference schemes to different questions and prove the convergence of all dif-ference schemes proposed in this article.Generalized Symmetric Regularized Long Wave equations and Rosenau-kdv equation are one of the most important nonlinear evolution equation, which has a rich physical background and content. Meanwhile,it has been wide-ly used in various different fields.For nonlinear evolution equation,it is necessary to study the numerical solutions of these equations.This report includes four chapters.The first chapter is an introduction. The research background and current situation of the problem are briefly introduced, some denotations and lemmas and research results are described.In the second chapter, we consider the following initial boundary value of the Gen-eralized Symmetric Regularized Long Wave equation ut+ρx+((up+1/p+1))x-uxxt=0, ∈(xL,xR),t∈(0,T)], ρt+ux=0, x∈(xL,xR),t∈=(0,T), u(x,0)-u0(x),ρ(x,0)ρ0(x),x∈[xL,xR], u(x,0) = u(xR,t) = 0, ρ(xL,t) = ρ(xR, t) = 0, t∈ (0, T] , where p ≥ 2 is positive integer.Using matrix knowledge,a compact difference scheme is proposed.At the same time, we prove the convergence of these difference schemes in the discrete L∞ norm for un, and in the discrete L2 norm for ρn. Thro ugh numerical results we can see that the order of convergence is O(τ2 + h4).In the third chapter,we consider the following initial boundary value of the Rosenau-kdv equation ut + uxxxxt + uxxx +ux + uux = 0 , (x,t) ∈ R× (0,T], u(x,0) = u0(x),x∈R , u(x + L,t) =u(x,t),(x,t)∈ R×[0,T]. this paper,Using weight coefficient,a decoupled two-level and nonlinear conservative fi-nite difference scheme with fourth-order accuracy is proposed to solve the Rosenau-kdv equation.By Brouwer-type theorem, we prove the existence of the solutions and the un- conditional convergence of this scheme in the discrete L^. norm.Through numerical results we can see that the order of convergence is O(τ~2+h~4).In the fourth chapter, we continue to study initial boundary value of the Rosenau-kdv equation. We give a three-level and linear conservative finite difference scheme with fourth-order accuracy.By Brouwer-type theorem, we prove the existence of the solutions. The unconditional convergence of this scheme in the discrete L∞ norm is also proved.Through numerical results we can see that the order of convergence is O(τ~2+h~4).