Zhendong Luo

Reduced-Order Modeling of the Upper Tropical Pacific Ocean Model using Proper Orthogonal Decomposition

The proper orthogonal decomposition (POD) is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations. In this paper, we make an initial effort to investigate problems related to POD reduced modeling of a large- scale upper ocean circulation in the tropic Pacific domain. We construct different POD models with different choices of snapshots and different number of POD basis functions. The results from these different POD models are compared with that of the original model. The main findings are: (1) the large-scale seasonal variability of the tropic Pacific obtained by the original model is well captured by a low dimensional system of order 22, which is constructed using 20 snapshots and 7 leading POD basis functions. (2) the RMS errors for the upper ocean layer thickness of the POD model of order 22 are less than 1m that is less than 1% of the average thickness and the correlation between the upper ocean layer thickness with that from the POD model is around 0.99. (3) Retaining modes that capture 99% energy is necessary in order to construct POD models yielding a high accuracy.

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.

A Quasi-Three-Dimensional Variably Saturated Groundwater Flow Model for Climate Modeling

In this study, a quasi-three-dimensional, variably saturated groundwater flow model was developed by approximately dividing the three-dimensional soil water and groundwater flow into an unsaturated vertical soil waterflow and a horizontal groundwaterflow to simulate the interactions among soil water, groundwater, and vegetation. The developed model consists of a one-dimensional unsaturated soil water flow model with the water table as the moving boundary using an adaptive grid structure for a vertical soil column formed based on discrete grid cells in a horizontal domain, a two-dimensional groundwater flow model for the horizontal domain, and an interface model connecting the two components for the horizontal grid cells in the domain. Synthetic experiments by the model were conducted to test the sensitivities of the model parameters of river elevation, ground surface hydraulic conductivity, and surface flux, and the results from the experimentsshowedtherobustnessoftheproposedmodelunderdifferentconditions.Comparisonofthesimulation by the model and that by a full three-dimensional scheme showed its feasibility and efficiency. A case of stream water conveyance in the lower reaches of the Tarim River was then applied to validate the developed model for simulation of the water table elevations at the Yingsu section. Finally, a numerical experiment by themodelfortheTarimRiverbasinwasconductedtodiscussthegroundwaterlatentflowforlarge-scalehighrelieftopographywithstreamwaterconveyance. Theresultsshowthatthemodel cansimulatethewatertable reasonably well.

A reduced-order extrapolation central difference scheme based on POD for two-dimensional fourth-order hyperbolic equations

The reduced-order extrapolating FD schemes are just developed by Luos research team Since 2013 (see 47-52) and a very new technique.The reduced-order extrapolation central difference scheme based on POD for two-dimensional fourth-order hyperbolic equations is first built by us.We provide theoretical analysis and examples to show the advantage of our method. This paper is concerned with establishing the reduced-order extrapolation central difference (ROECD) scheme based on proper orthogonal decomposition (POD) for two-dimensional (2D) fourth-order hyperbolic equations. For this purpose, we first develop the classical central difference (CD) scheme for the 2D fourth-order hyperbolic equations and analyze its stability and convergence. Then by making use of the POD method, we build the ROECD scheme with fewer degrees of freedom and sufficiently high accuracy and furnish the error estimates of the ROECD solutions and the algorithm procedure for solving the ROECD scheme. Finally, we employ some numerical examples to confirm the correctness of theoretical conclusions. This implies that ROECD scheme is feasible and efficient for seeking the numerical solutions of the 2D fourth-order hyperbolic equations.

A reduced-order Crank-Nicolson finite volume element formulation based on POD method for parabolic equations

In this paper, a proper orthogonal decomposition (POD) method is used to deal with a classical Crank-Nicolson finite volume element (CNFVE) method for two-dimensional parabolic equations. A reduced-order CNFVE formulation with lower dimensions and sufficiently high accuracy based on POD technique is established, the error estimates between reduced-order CNFVE solutions based on the POD method and classical CNFVE solutions are provided, and the extrapolation algorithm for solving reduced-order CNFVE formulation is implemented. Some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions. Moreover, it is shown that the reduced-order CNFVE formulation based on POD method is feasible and efficient for solving two-dimensional parabolic equations.

A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine the classical finite volume element (FVE) method with the POD method to obtain a reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional solute transport problems, which have real life practical applications. We then provide error estimates between the reduced-order POD FVE solutions and classical FVE solutions and we provide implementation of an extrapolation algorithm for solving the reduced-order FVE formulation. Thus, we provide the theoretical basis for practical applications. A numerical example is then used to ascertain that the results of numerical computation are consistent with the theoretical derivations. Moreover, it is shown that the reduced-order FVE formulation based on POD method is both feasible and efficient for solving two-dimensional solute transport problems.

A reduced-order extrapolation algorithm based on SFVE method and POD technique for non-stationary Stokes equations

In this paper, a reduced-order extrapolation algorithm (ROEA) based on proper orthogonal decomposition (POD) technique and classical stabilized finite volume element (SFVE) method for non-stationary Stokes equations is established. The error estimates between the ROEA solutions and the classical SFVE solutions and the implementation for solving ROEA are provided. Some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that ROEA based on SFVE formulation and POD method is feasible and efficient for solving non-stationary Stokes equations.

A POD-based reduced-order Crank–Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations

Abstract Based on proper orthogonal decomposition (POD), a new type of reduced-order Crank–Nicolson finite volume element extrapolating algorithm (CNFVEEA) including very few degrees of freedom but holding fully second-order accuracy for two-dimensional (2D) Sobolev equations is established firstly. Then, the error estimates of POD-based reduced-order CNFVEEA solutions are provided, which acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the procedure for the implementation of the POD-based reduced-order CNFVEEA is given. Finally, a numerical example is presented illustrating that the numerical computational conclusions are consistent with theoretical ones. Moreover, it is shown that the POD-based reduced-order CNFVEEA is very suitable to finding numerical solutions of 2D Sobolev equations and is better than the POD-based FVE formulation with first-order accuracy in time.

A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation

In this study, we devote ourselves to the reduced-order extrapolated finite difference iterative (ROEFDI) modeling and analysis for the two-dimensional (2D) Sobolev equation. To this end, we first establish the reduced-order extrapolated finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the 2D Sobolev equation via the proper orthogonal decomposition (POD) technique. And then, we analyze the stability and convergence of the ROEFDI solutions. Finally, we use the numerical experiments to verify the feasibility and effectiveness of the ROEFDI scheme.

Analysis of a space–time continuous Galerkin method for convection-dominated Sobolev equations ☆

Abstract The convergence of space–time continuous Galerkin (STCG) method for the Sobolev equations with convection-dominated terms is studied in this article. It allows variable time steps and the change of the spatial mesh from one time interval to the next, which can make this method suitable for numerical simulations on unstructured grids. We prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in L ∞ ( H 1 ) norm which do not require any restriction assumptions on the space and time mesh size. Finally, some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis.