Numerical Methods For Some Fourth Order Nonlinear Partial Differential Equations

As mathematical models,high order nonlinear parabolic equations describe many phenomena which exist in physics;chemistry,geography,environmental science and many other fields.They play important roles in nonlinear science.In this thesis,we consider the numerical methods for three types of fourth order nonlinear partial dif-ferential equations with extensive application background.We study the priori bounds and convergence in theory,and specifically prove the priori bounds with H2 norm.The H2 norm priori bounds of numerical solutions is the basis of error estimates for fourth order equations.Numerical experiments are also presented to demonstrate the theoretical results.Firstly,we consider the fourth order nonlinear parabolic equation describing the growing process of crystal surface A few researches have been done about numerical methods in the study of this equa-tion.We published a paper about the finite element method.Then we consider the finite difference method because of the numerical calculations of cubic finite element method bringing large computation costs.This thesis study the linearized difference scheme in one-dimensional problem and the Crank-Nicolson difference scheme in two-dimensional problem.We prove the priori bounds and convergence,and present numer-ical experiments.The nonlinear term is a rational function which is very complex if we handle it directly,while the complexity will reduce with integration by parts.We make theoretical analysis by continuous inner product in the study of finite element method.The nonlinear term can be handled with integration by parts.We give a semi-implicit difference scheme without the cost of computing for one-dimensional problem,because of the nonlinear term can't meet the needs of integration by parts in the simple transplant difference scheme.We can handle the nonlinear term with the same way as the study of the finite element method.The time accuracy is first order and the space accuracy is second order in theory,and the experimental results conform it.We also establish a Crank-Nicolson difference scheme for two-dimensional problem and define discrete in-ner product that matches it,so that we can deal with the nonlinear term with the similar approach.Compared to the linearized difference scheme,although the Crank-Nicolson difference scheme needs more computation,the convergence accuracy reaches second order in time.The experimental results are consistent with the theoretical results.Secondly,we study the pseudo-spectral method for Extended Fisher-Kolmogorov(EFK)equation There are some research results about the numerical methods for EFK equation.These methods involve kinds of finite element methods and finite difference methods.In this thesis,we give theoretical analysis and numerical simulation about Fourier pseudo-spectral method.A priori bounds for the solution of semi-discrete scheme is proved,and a fully discrete scheme based on Crank-Nicolson method is derived.Further,we consider the priori bounds of the fully discrete numerical solution and establish the error estimates in L2 norm.In particular,the solution space and the discrete inner product not only match with boundary conditions but also maintain the orthogonality of basis.According to the characteristics of nonlinear term,we introduce the following energy function where H(uh)h=1/4(1 uh2z)2,and H'(uh)= f(uh).We get a priori bounds by the property of the energy function.Similarly,to obtain the stability results of the fully discrete solution,we introduce the discrete energy function.Furthermore,we can get the convergence order which is similar to the Fourier spectral method.We also carry out some numerical experiments to support our theoretical claims.Finally,we discuss the long time behavior of the numerical solution for the con-vective Cahn-Hilliard equation where f(u)is a quartic polynomial form,and g(7)is a quintic polynomial form.We present a Fourier spectral method for solving the convective Cahn-Hilliard equation.In this thesis,we study the long time behavior of the numerical solution and consider the priori bounds numerical.Error estimates with limited time are made.It is difficult to get the priori bounds with H0 norm because of the particularity of ?g(u).We build an antisymmetric scheme against g(u),and establish the Fourier spectral method by it.With the antisymmetric scheme,the effects of g(u)are excluded in the proof of priori bounds with H0 norm.Then,we get the long time priori bounds with H0 norm.We usually prove the priori bounds with H2 norm by energy function,but we hardly find the energy function which matches with H2 norm for this problems.So we mainly use the priori bounds with H0 norm and Nirenberge's inequality,and make the best use of the forth order term,gaining good theoretical results.