KPOD Model Order Reduction Methods For Partial Differential Equations With Variable Coefficients

Many problems in the engineering field can be modeled by partial differential equations.By solving these partial differential equations,relatively accurate data can be provided for engineering design,so as to ensure the safe and efficient operation of engineering projects.However,the scale of the ordinary differential equations obtained by space discretization of these partial differential equations is very large,and solving them requires a great amount of computation and time cost.The model order reduction(MOR)method can effectively approximate the large-scale system by generating a small-scale system,thus it is an effective method to solve this kind of problem.In this thesis,the KPOD(Krylov enhanced Proper Orthogonal Decomposition)methods are studied for the partial differential equations with variable coefficients.The specific contents are as follow:In the first part,the Galerkin KPOD MOR method for a partial differential equation with variable coefficients is studied.First of all,based on the variational theory of Galerkin finite element,the spatial discrete scheme of the partial differential equation with variable coefficients is established,and the system of differential equations with time-varying coefficients is obtained,i.e.,the time-varying system.Then,the coefficient matrix of the time-varying system is discrete,and the input Krylov subspace based on block Arnoldi algorithm is constructed.The error bound between the Galerkin finite element solution and the Galerkin KPOD solution is given.Finally,a numerical example is given to verify the feasibility of the proposed algorithm for solving partial differential equations with variable coefficients.In the second part,the two-sided KPOD MOR methods for a partial differential equation with variable coefficients are studied.Firstly,the KPOD MOR method based on the two-sided Krylov subspace is proposed.For the time-varying systems obtained,a two-sided Krylov subspace based on block Arnoldi and block Lanczos processes are constructed,and the projection matrices are obtained.The transformation matrices are obtained by POD method,and the moment matching characteristics between the transfer function of the projection system and that of the original system are analyzed.Further,the two-sided KPOD MOR method based on Laguerre polynomials in frequency domain is proposed.By constructing the two-sided Krylov subspace based on Laguerre polynomials,the projection matrices of the subspace are obtained,and the transformation matrices are obtained by POD method.The moment matching characteristics of the transfer function between the projection system and the original system are analyzed,and the error bounds between the finite element solution and the reduced solution are given.A numerical example is given to verify the feasibility of the proposed algorithms for solving partial differential equations with variable coefficients.