The Numerical Solution Of Generalized KdV-RLW-Rosenau Equations Based On POD Method

L∞-error estimates for B-spline Galerkin finite element(GFE)solution of the reduced order modeling for generalized Korteweg-de Vries-regularized-long wave-Rosenau(Kd V-RLW-Rosenau)equation are considered.The semidiscrete B-Spline Galerkin scheme is studied using appropriate projections.For fully discrete B-spline Galerkin scheme,we consider the Crank-Nicolson method and analyze the corresponding error estimates in time.A proper orthogonal decomposition(POD)method is applied to a GFE formulation for generalized Kd V-RLW-Rosenau equation such that it is reduced into a POD GFE formulation with lower dimensions and enough high accuracy.Numerical solutions of nonlinear complex systems are expensive with respective to both storage and CPU costs.It is impossible to calculate the optimal control or feedback control of complex systems in real time Even with the use of good mesh generators,discretization schemes and solution algorithms for the computational simulation of complex,turbulent and chaotic systems still remain a formidable endeavor.In order to solve these problems,dimensionality reduction methods are proposed.Among them,the proper orthogonal decomposition(POD)method is an effective dimensionality reduction method.The POD method is an effective numerical method that can not only ensure the calculation accuracy,but also greatly reduce the number of unknowns in the numerical calculation process.This dimensionality reduction method has been widely used in the fields of signal processing,geophysics and statistics.In this paper,the POD method is used to analyze the numerical solution problem of the dimensionality reduction model of the generalized Kd V-RLW-Rosenau equation.In fact,POD combined with Galerkin projection has been used to formulate dimensionality reduction models of dynamic systems for many years.The research of this paper is as follows: In the first chapter,the research background and the introduction of the equations are given.In the second chapter,the weak form(2.1)of the equation(1.1)is obtained by means of integration by parts,The existence and uniqueness of the solution of(2.1)are proved by the continuation theorem,Gronwall inequality,Sobolev’s inequality and Young’s inequality.The finite element approximate solutions are discussed,and theoretically derives the second-order spatial error estimate in the norm using methods such as auxiliary projection and Cauchy-Schwartz inequality.A fully discrete scheme using the Crank-Nicolson method for time is proposed in the third Chapter,the existence and uniqueness of the solution of the fully discrete scheme(3.1)is proved by Brouwer’s fixed point theorem,and the error of the Galerkin-Crank-Nicolson scheme is deduced using elliptical projection.The second order accuracy in time is proved at the same time.The fourth chapter introduces the generation of POD basis and the construction technology of reduce order model,and uses POD GFE method to deduce the error estimation of reduce order model.some numerical experiments are given to demonstrate the effectiveness and accuracy of the proposed method.The error between the reduced POD GFE solution and the usual GFE solution are analysed.It is shown that the result of numerical computation is consistent with theoretical conclusions,this validates the feasibility and efficiency of the POD method.In the fifth Chapter,feedback control of equations is considered using a linear quadratic regulator design.