Model Order Reduction Methods For Parabolic Partial Differential Equations With Variable Coefficients Based On Discrete Polynomials

Those partial differential equations play an increasingly important role in society.In general,many models in complex fields such as physics and engineering are created by those equations to describe their relevant features.The numerical calculation of this kind of equations has always been a hot topic for scholars.Up to now,solving those partial differential equations,which is usually discreted by using time and spatial discretization,and which leads to the number of discrete equations is very large,and the fast and reliable calculation is challenging in many cases.In this case,the model order reduction method provides an effective solution.Since those coefficients of those equations are related to time and space,it is more difficult to solve them.Therefore,it is necessary to study the model order reduction method of those partial differential equations with variable coefficients.In this thesis,we study the model order reduction method for parabolic partial differential equations with variable coefficient based on discrete polynomials.The specific contents are showed as follows:In the first part,we study the time domain model order reduction methods based on the discrete polynomials.Firstly,based on Galerkin variational theory and finite difference method,we establish a discrete scheme of the variable coefficient parabolic partial differential equation,and obtain the discrete system.Secondly,we discuss the one-sided time domain model order reduction method based on discrete Laguerre polynomial for the discrete system.In the space,which is spanned by the time discrete Laguerre polynomial,the transformation matrix is obtained by calculating a group of expansion coefficients of the corresponding solution vector,and then the reduced system is constructed.The invariant coefficient property of the output variables between the original system and the reduced system is analyzed,and the error bound is given.Further,we discuss the two-sided time domain model order reduction method based on the discrete Laguerre polynomial and Charlier polynomial.Those expansion coefficients are calculated by discrete Laguerre polynomial and Charlier polynomial,we construct the left transformation matrix and the right transformation matrix,and then construct the reduced system in the Petrov-Galerkin projection frame.The coefficient invariant propertie of the output variables between the original and reduced systems is analyzed.Finally,the feasibility of the proposed methods for solving parabolic variable coefficient partial differrential equation is verified by numerical examples.In the second part,we study the frequency domain model order reduction methods of based on discrete Laguerre polynomial.Firstly,based on Galerkin variational theory and finite difference method,we establish a discrete scheme of the variable coefficient parabolic partial differential equation,and obtain the discrete system.Secondly,we discuss the one-sided frequency domain model order reduction method based on the discrete Laguerre polynomial.By constructing input Krylov subspaces based on discrete Laguerre polynomialsl,and utilizing the block Arnoldi algorithm,we construct the reduced system.The moment-matching propertie of the transfer functions between the original and reduced system is analyzed,and the error bound is given.Further,by constructing the output Krylov subspace based on the discrete Laguerre polynomial in the frequency domain,we discuss the two-sided frequency domain model order reduction method based on the discrete Laguerre polynomial,and analyze the moment-matching propertie of the transfer function between the original and the reduced system.Finally,the feasibility of the proposed methods for solving parabolic variable coefficient partial differrential equation is verified by numerical examples.