Research Of Spectral Methods Based On POD Reduced-Order Extrapolation Algorithm

This paper mainly focuses on studying the spectral method based on The proper orthogonal decomposition(POD)reduced-order extrapolation algorithm.It is well known that the numerical methods for partial difference equations(PDEs)mainly includes finite difference method,finite element method and spectral method.By comparison with the two methods,the spectral method has superior accuracy so that it has been attracted by many researchers.Although the spectral method is viable in theory to find the numerical solution of PDEs,it still includes many unknowns for a large engineering problem.It affects not only on the calculation precision,but also does the efficiency.Therefore,it is a very significant issue how to build a modified spectral method with fewer degrees of freedom under guaranteeing accurary of numerical solutions.The proper orthogonal decomposition(POD)method is a resultful way to guarantee calculation accurary and reduce the freedom degrees in numerical computation,and this method has been intensively applied in the numerical calculation for PDEs by researchers.Therefore,in this paper,we focuse on the research of the spectral method based on POD reduced-order extrapolation algorithm,mainly set up some reduced-order spectral methods with more calculated accurary and higher calculation efficiency.The contents of this paper are organized as follows:Part 1,we,respectively,use the classical Galerkin spectral method and the POD-based reduce-order extrapolation Galerkin spectral method to deal with the 2D second-order hyperbolic equations.We will first use the classical spectral method to deal with this equation,bulid the iterative scheme for the solutions of the classical spectral method,offer its convergence and stability analysis.And then,by formulating POD bases,we develop the POD-based reduced-order extrapolation spectral method,analyze its convergence,offer the implementations and give some numerical simulations to validate the feasibility and availability of POD-based reduce-order extrapolation Galerkin spectral method.Part 2,we,respectively,apply the classical collocation spectral method and the POD-based reduce-order extrapolation collocation spectral method to deal with the 2D Sobolev equations.We first write the Sobolev equations as an equivalent form of variation by using Green's formula,and show that there exists a unique solution of this variational form.And then,we discretize the variational form by Legendre-Gauss-Lobatto collocation points in space,and set up the classical spectral method iterative scheme for the 2D Sobelev equations.Next,we analyze the existence,uniqueness,stability and convergence of solution.On this basis,we produce a set of POD bases by the snapshots which get from the initial few classical spectral solutions,establish the POD-based spectral method iterative scheme,discuss the existence,uniqueness,stability,and convergence of the scheme solutions.Moerover,we use some numerical experiments to show that the POD-based spectral method is superior to the classical one and to verify that the results of numerical computations are accorded with the theoretical analysis and that both methods are very efficient for solving the 2D Sobolev equations.Part 3,we use the Crank-Nicolson collocation spectral method and POD-based reduced-order extrapolate Crank-Nicolson collocation spectral method to deal with the 2D viscoelastic wave equations.We first write the viscoelastic wave equations as variational form by using Green's formula,establish the Crank-Nicolson collocation spectral method scheme for 2D viscoelastic wave equations,and discuss the existence,uniqueness,stability,convergence of the scheme solutions.And then,using POD algorithm to reduce the order of the coefficient vectors of the solutions for the classical spectral method of 2D viscoelastic wave equations,we establish a reduce-order extrapolation Crank-Nicolson collocation spectral method scheme so that the reduce-order scheme has the same basis functions as the classical spectral method scheme and maintains all advantages of the classical spectral method scheme.Then,by means of matrix analysis,we discuss the existence,stability,and convergence of the reduce-order spectral method scheme solutions.Finally,we give some numerical experiments to validate that the numerical computational results are consistent with the theoretical ones such that the effectiveness and feasibility of the Crank-Nicolson collocation spectral method and reduce-order extrapolation Crank-Nicolson collocation spectral method for 2D viscoelastic wave equations are further verified.Part 4,we mainly focus on a collocation spectral-projection method for 2D unsteady Stokes equations.First,we use a projection method with pressure-correction way to discrete the unsteady Stokes equations in time,and apply the spectral method to approximate the solution of this equation in spatial variables such that we establish the collocation spectral-projection method scheme for the 2D unsteady Stokes equations.Then,we discuss the existence,stability,and convergence for the solutions of the classical collocation spectral-projection method scheme.Finally,by using of a specific numerical example,the correctness of theoretical analysis can be verified,which showed that the classical collocation spectral-projection method is feasible for solving the 2D unsteady Stokes equations.