The Study On Highly Efficient Difference Scheme For Benjamin-Bona-Mahony-Burgers' Equation

Aiming at Benjamin-Bona-Mahony-Burgers'(BBMB)equation in the paper,we construct three type difference schemes and the corresponding theoretical analysis of the numerical schemes has been discussed.Finally,numerical examples verify the numerical theoretical results.This paper mainly consists of the following four parts.In Chapter 1,we introduce the background,development process and current research progress of BBMB equations and the notation required for the research of this paper are shown.In Chapter 2,we consider the one-dimensional BBMB equation with Dirichlet boundary,for which we construct two linearized difference schemes and perform theoretical analysis and numerical experiments.For the construction of the two-level scheme,the nonlinear term is linearized via averaging k and k+1 floor.We prove the unique solvability of difference scheme in detail with the second-order convergence in time and space.For the three-level linearized scheme,the extrapolation technique is utilized to linearize the nonlinear term based on? function.We obtain the conservation,boundedness,unique solvability and convergence of numerical solutions with the convergence order O(?~2+h~2)at length.Furthermore,we also extend our work to obtain two types of Newtonian linearized finite difference schemes for the BBMB equation with nonlinear source term.The applicability and accuracy of both schemes are demonstrated by numerical experiments.In Chapter 3,for the one-dimensional BBMB equation with periodic boundary,we employ a novel three-point fourth-order compact operator in space to establish an efficient compact difference scheme.The detailed derivation is carried out based on the reduction order method together with a three-level linearized technique.The conservative invariant,boundedness and unique solvability are studied at length.The convergence is proved by the technical energy argument and induction method with the optimal convergence order O(?~2+h~4)in the sense of the maximum norm.Based on the consistent boundedness of the numerical solution,we derive the unconditional stability of the numerical solution.The present scheme is very efficient in practical computation since only a linear system of equations with symmetric cyclic matrices as coefficients needs to be solved at each time step.The extensive numerical examples verify our theoretical results and demonstrate the scheme's superiority when compared with state-ofthe-art those in the references.In last Chapter,the research content and innovation points of this paper are summarized and explained,and the next research direction is pointed out.