B-method For Numerical Solutions Of Some Nonlinear Parabolic Equations

B-method is an efficient numerical method recently developed for nonlinear parabolic partial differential equations with blow-up solutions.This method was firstly proposed by Beck et al.[13]in 2015 to find numerical solutions to second order nonlinear parabolic equations with blow-up phenomena,and therefore is named after the initial letter of the blow-up.Note that the solutions of nonlinear parabolic equations have large rate of change and large value near the blow-up time,both of them are dominated over the term of spatial derivatives.Based on the technique of the variation of constants,B-method first solve exactly the nonlinear ODEs,which determine largely the blow-up phenomena in the whole system,and then introduce the term including spatial derivatives to the numerical schemes by means of the technique of the variation of constants,since the term of spatial derivatives only affects the blow-up behaviour slightly.B-method fully consid-ers the geometric properties of solutions in the design process,and can be regarded as an extension and development of the classical time discretization schemes.Meanwhile,B-method can be viewed as an approach of preserving a special geometry structure of solutions.In this thesis,we apply B-method to solve fourth order nonlinear parabolic equations and second order convective reaction-diffusion equations with blow-up solutions.We also extend B-method to the second order parabolic equations with quenching solutions.For each specific problem,we derive B-method schemes and perform truncation error anal-ysis,prove the existence and uniqueness of numerical solutions and carry out numerical experiments.This thesis is divided into five chapters:In Chapter 1,we firstly summarize the preliminaries of nonlinear parabolic equa-tions,and then briefly introduce the history of B-method.Secondly,we briefly describe three models we concerned and their recent research results.Lastly,for an example of a very general equation,we show how to construct B-method schemes.In Chapter 2,we apply B-method on a class of fourth order nonlinear parabolic equations with blow-up solutions.Firstly we introduce several numerical schemes of the concerned equation,and then take one of the schemes VCFE as an example to derive local truncation errors of B-method scheme and its corresponding Forward Euler method scheme,by comparison,we draw a conclusion that the local truncation error of B-method scheme is smaller.Secondly,we transform one of the numerical schemes of B-method to a fourth order elliptic equation and prove the existence and uniqueness of the elliptic equation by means of upper and lower solutions theory,which means the existence and uniqueness of numerical solution are proved.Lastly,three numerical examples are pro-vided to verify the errors of B-method are smaller than the errors of the corresponding classical method.In Chapter 3,B-method is utilized to solve second order convective reaction-diffusion equations with blow-up solutions.Firstly we introduce the numerical schemes of the equation,and then prove that the truncation errors of B-method are smaller than the corresponding classical method.Secondly,we give the proof of the existence of a nu-merical solution under some reasonable assumptions.Lastly,we render three examples to verify the errors of B-method are smaller than the classical method using the same discretization.Chapter 4 is devoted to numerical analysis of B-method for solving second order nonlinear parabolic equations with quenching solutions.Firstly,we solve the nonlinear part of the equations and introduce the linear part by means of the variation of constants technique,and therefore,derive the B-method numerical schemes.Secondly,we take one of these schemes as an example to prove the existence of the numerical solution.Finally,one example is supplied to estimate the numerical quenching time,and illustrate our theoretical resultsIn Chapter 5,we present a summary of this thesis.