A reduced-order finite difference extrapolation algorithm based on POD technique for the non-stationary Navier–Stokes equations

In this paper, a proper orthogonal decomposition (POD) technique is used to establish a reduced-order finite difference (FD) extrapolation algorithm with lower dimensions and sufficiently high accuracy for the non-stationary Navier–Stokes equations, and the error estimates between the reduced-order FD solutions and the classical FD solutions and the implementation for solving the reduced-order FD extrapolation algorithm are provided. Two numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order FD extrapolation algorithm based on POD method is feasible and efficient for solving the non-stationary Navier–Stokes equations.     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

A pod reduced-order spdmfe extrapolating algorithm for hyperbolic equations

In this article, a proper orthogonal decomposition (POD) method is used to study a classical splitting positive definite mixed finite element (SPDMFE) formulation for second-order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.     

非饱和土壤水流问题基于POD 方法的降阶有限体积元格式及外推算法实现

利用特征投影分解(proper orthogonal decomposition, 简记为POD) 方法对非饱和土壤水流问题的经典有限体积元格式做降阶处理, 建立一种具有足够高精度维数较低的降阶有限体积元格式, 并给出这种降阶有限体积元解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的. 进一步表明了基于POD 方法的降阶有限体积元格式对求解非饱和土壤水流问题数值解是可靠和有效的.     

Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.     

A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR HYPERBOLIC EQUATIONS

In this article, a proper orthogonal decomposition （POD） method is used to study a classical splitting positive definite mixed finite element （SPDMFE） formulation for second- order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.     

二维土壤溶质输运方程基于POD方法的降阶CN有限体积元外推算法

利用Crank-Nicolson(CN)有限体积元方法和特征投影分解方法建立二维土壤溶质输运方程的一种维数很低、精度足够高的降阶CN有限体积元外推算法,并给出这种外推算法的降阶CN有限体积元解的误差估计和算法的实现.最后用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶CN有限体积元外推算法的优越性.     

交通流模型基于特征投影分解技术的外推降维有限差分格式

利用特征正交分解(proper orthogonal decomposition,简记为POD)技术研究交通流的Aw-Rascle-Zhang(ARZ)模型. 建立一种基于 POD方法维数较低的外推降维有限差分格式, 并用数值例子检验数值计算结果与理论结果相吻合, 进一步表明基于POD方法的外推降维有限差分格式对于求解交通流方程数值解是可行和有效的.     

非定常Burgers方程基于POD方法的全二阶差分格式降阶外推算法

利用特征投影分解方法和奇值分解技术对非定常Burgers 方程的经典全二阶有限差分格式做降阶处理, 给出一种时间和空间变量都是二阶精度的降阶差分格式, 并给出这种降阶全二阶精度差分解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的.     

A reduced-order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two-dimensional viscoelastic wave equations

In order to reduce the order of the coefficient vectors of the solutions for the classical Crank–Nicolson collocation spectral (CNCS) method of two-dimensional (2D) viscoelastic wave equations via proper orthogonal decomposition, we first establish a reduced-order extrapolated CNCS (ROECNCS) method of the 2D viscoelastic wave equations so that the ROECNCS method has the same basis functions as the classical CNCS method and maintains all advantages of the classical CNCS method. Then, by means of matrix analysis, we discuss the existence, stability, and convergence for the ROECNCS solutions so that the theory analysis becomes very concise. Finally, we utilize some numerical experiments to validate that the consequences of numerical computations are accordant with the theoretical analysis such that the effectiveness and viability of the ROECNCS method are further verified. Therefore, both theory and method of this paper are new and totally different from the existing reduced-order methods.     

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

We present some results on the stabilization of reduced-order models (ROMs) for thermal fluids. The stabilization is achieved using robust Lyapunov control theory to design a new closure model that is robust to parametric uncertainties. Furthermore, the free parameters in the proposed ROM stabilization method are optimized using a data-driven multi-parametric extremum seeking (MES) algorithm. The 2D and 3D Boussinesq equations provide challenging numerical test cases that are used to demonstrate the advantages of the proposed method.     

A reduced FE formulation based on POD method for hyperbolic equations

A proper orthogonal decomposition (POD) method was successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, namely, apply POD method to a classical finite element (FE) formulation for second-order hyperbolic equations with real practical applied background, establish a reduced FE formulation with lower dimensions and high enough accuracy, and provide the error estimates between the reduced FE solutions and the classical FE solutions and the implementation of algorithm for solving reduced FE formulation so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced FE formulation based on POD method is feasible and efficient for solving FE formulation for second-order hyperbolic equations.     

ASYMPTOTIC ERROR EXPANSION AND DEFECT CORRECTION FOR SOBOLEV AND VISCOELASTICITY TYPE EQUATIONS

1.IntroductionLetObearectangulardomain.WeareconcernedwiththeRichardsonextrapolationanddefectcorrectionofthefiniteelementapproximationstothesolutionsofthefollowingsimpleSobolevtypeequationandviscoelasticitytypeequationTheextrapolationtechniqueusedforthefin…     

Application of Richardson extrapolation to the numerical solution of partial differential equations

Richardson extrapolation is a methodology for improving the order of accuracy of numerical solutions that involve the use of a discretization size h. By combining the results from numerical solutions using a sequence of related discretization sizes, the leading order error terms can be methodically removed, resulting in higher order accurate results. Richardson extrapolation is commonly used within the numerical approximation of partial differential equations to improve certain predictive quantities such as the drag or lift of an airfoil, once these quantities are calculated on a sequence of meshes, but it is not widely used to determine the numerical solution of partial differential equations. Within this article, Richardson extrapolation is applied directly to the solution algorithm used within existing numerical solvers of partial differential equations to increase the order of accuracy of the numerical result without referring to the details of the methodology or its implementation within the numerical code. Only the order of accuracy of the existing solver and certain interpolations required to pass information between the mesh levels are needed to improve the order of accuracy and the overall solution accuracy. Using the proposed methodology, Richardson extrapolation is used to increase the order of accuracy of numerical solutions of the linear heat and wave equations and of the nonlinear St. Venant equations in one‐dimension. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

守恒高阶各向异性交通流模型基于POD方法的降阶外推差分格式

利用Godunov流方法和特征投影分解方法,对守恒高阶各向异性交通流模型建立一种自由度很少、精度足够高的降阶外推差分算法, 并给出这种降阶外推差分算法近似解的误差估计和算法实现.最后,用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶外推差分算法的优越性.     

Numerical solution of the heat equation by infinite-order ordinary differential equations

A numerical algorithm is given for solving a class of infinite-order differential equations previously discussed by the authors. These equations yield solutions to certain time-dependent boundary-value problems for the heat equation. An extrapolation process is given for increasing the speed of convergence of the sequence generated by the method. © 1992 John Wiley & Sons, Inc.     

Sobolev方程的全离散有限体积元格式及数值模拟

本文研究二维Sobolev方程的有限体积元方法, 给出一种全离散化有限体积元格式及其有限体积元解的误差估计,并用数值例子说明数值计算的结果与理论结果是相吻合的, 进一步说明了有限体积元方法比其他数值方法更优越.     

A multistep collocation method for solving advection equations with delay

Advection equations with delay are appeared in the modeling of the dynamics of structured cell populations. In this article, we construct an efficient two-dimensional multistep collocation method for the numerical solution of a class of advection equations with delay. Equations with aftereffect and equations with both aftereffect and retardation of a state variable are considered. Computability of the algorithm and convergence properties of the proposed numerical method are analyzed for solutions in appropriate Sobolev spaces, and it is shown that the proposed scheme enjoys the spectral accuracy. Numerical examples are given and comparison with other existing methods in the literature is made to demonstrate the efficiency, superiority and high accuracy of the presented method.     

A reduced-order extrapolating space–time continuous finite element method for the 2D Sobolev equation

This paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two-dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper orthogonal decomposition to develop a reduced-order extrapolating STCFE (ROESTCFE) vector model with the lower dimension, and demonstrate the existence, stability, and convergence for the ROESTCFE solutions by the matrix means, resulting in the very concise and flexible theoretical analysis. Lastly, we examine the effectiveness of the developed ROESTCFE model by several numerical tests. It is shown that the ROESTCFE method is computationally very cheap in actual applications.