A reduced finite element formulation based on POD method for two-dimensional solute transport problems

Luo, Zhendong; Li, Hong; Zhou, Yanjie; Xie, Zhenghui

doi:10.1016/j.jmaa.2011.06.051

Cited by 57 publications

(23 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The results also show the convergence with respect to PRS order. It is interesting to note that the relative errors of the PRS solutions are similar to those of the reduced‐order model of the reaction‐advection‐diffusion equation found in the work of McLaughlin et al and of the solute transport problem found in the work of Luo et al In addition to this, the convergence with respect to PRS order shows similarities to the convergence of the aforementioned reduced‐order models with respect to the number of basis functions employed.…”

Section: Investigation Of the Range Of Applicabilitysupporting

confidence: 76%

An indicator‐based problem reduction scheme for coupled reactive transport models

Freeman

Cleall

Jefferson

2019

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary A number of effective models have been developed for simulating chemical transport in porous media; however, when a reactive chemical problem comprises multiple species within a substantial domain for a long period of time, the computational cost can become prohibitively expensive. This issue is addressed here by proposing a new numerical procedure to reduce the number of transport equations to be solved. This new problem reduction scheme (PRS) uses a predictor‐corrector approach, which “predicts” the transport of a set of non‐indicator species using results from a set of indicator species before “correcting” the non‐indicator concentrations using a mass balance error measure. The full chemical transport model is described along with experimental validation. The PRS is then presented together with an investigation, based on a 16‐species reaction‐advection‐diffusion problem, which determines the range of applicability of different orders of the PRS. The results of a further study are presented, in which a set of PRS simulations is compared with those from full model predictions. The application of the scheme to the intermediate‐sized problems considered in the present study showed reductions of up to 82% in CPU time, with good levels of accuracy maintained.

show abstract

Section: Investigation Of the Range Of Applicabilitysupporting

confidence: 76%

An indicator‐based problem reduction scheme for coupled reactive transport models

Freeman

Cleall

Jefferson

2019

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…In fact, u n E y , H). Apparently, the inequality (25) implies that u n d u (u = E x , E y , H) are the optimal approximations of u n whose errors are no more than λ d u +1 . Thus, Φ u (u = E x , E y , H) are just three orthogonal optimal POD bases of A u (u = E x , E y , H).…”

Section: Formulate the Pod Basismentioning

confidence: 99%

“…It had widely been applied in the real-life numerical computations (see, e.g., [4,32,40]). Particularly, it had been used to build some POD reduced-order numerical computational methods for the partial differential equations such as the POD Galerkin reduced-order models (see, e.g., [1,3,13,14]), the POD finite element reduced-order models (see, e.g., [18,19,21,25]), and the POD finite volume element reduced-order models (see, e.g., [24,29]). The POD reduced-order models can not only ensure the sufficiently accurate numerical solutions, but also lessen the computational load and improve the calculating efficiency.…”

Section: Introductionmentioning

confidence: 99%

A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium

Luo

Gao

2016

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

This paper is concerned with establishing a reduced-order finite difference time-domain (FDTD) extrapolating scheme based on proper orthogonal decomposition (POD) method for the two-dimensional (2D) Maxwell equations in a lossy medium. For this purpose, we first review the classical FDTD scheme and associated results for the 2D Maxwell equations in a lossy medium. And then, we formulate three sets of POD bases by means of the solutions obtained from the classical FDTD scheme on the very short time interval, establish the POD reduced-order FDTD extrapolating scheme, and furnish the error estimates among the POD reduced-order FDTD solutions and the classical FDTD solutions as well as the accuracy solution for the 2D Maxwell equations in a lossy medium. Finally, we enumerate two numerical examples to confirm that the POD reduced-order FDTD extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium.

show abstract

“…It has been extensively used in analysis of signal together with pattern recognition (see [5]), statistical computation (see [9]), and computational fluid dynamics (see [29]). In recent years, it also has been successfully applied to the order-reduction for the Galerkin method (see, e.g., [10,11]), the finite element method (see, e.g., [18,22]), the FD scheme (see, e.g., [24,31]), finite volume element method (see, e.g., [21,23]), and reduced basis methods (see, e.g., [1,6,28]) for PDEs. However, the most existing POD reduced order methods (see, e.g., [1, 2, 5-7, 9-11, 18, 21-25, 28-31]) are built by the POD basis formed with the classical solutions at the all time nodes on [0, T ], before repetitively computing the reduced order solutions at the same time nodes.…”

Section: Introductionmentioning

confidence: 99%

A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Cao¹,

Luo²

2018

J. Nonlinear Sci. Appl.

Self Cite

View full text Add to dashboard Cite

This article mainly studies the order-reduction of the classical Crank-Nicolson finite difference (CNFD) scheme for the Riesz space fractional order differential equations (FODEs) with a nonlinear source function and delay on a bounded domain. For this reason, the classical CNFD scheme for the Riesz space FODE and the existence, stability, and convergence of the classical CNFD solutions are first recalled. And then, a reduced-order extrapolating CNFD (ROECNFD) scheme containing very few degrees of freedom but holding the fully second-order accuracy for the Riesz space FODEs is established by means of proper orthogonal decomposition and the existence, stability, and convergence of the ROECNFD solutions are discussed. Finally, some numerical experiments are presented to illustrate that the ROECNFD scheme is far superior to the classical CNFD one and to verify the correctness of theoretical results. This indicates that the ROECNFD scheme is very effective for solving the Riesz space FODEs with a nonlinear source function and delay.Keywords: Crank-Nicolson finite difference scheme, Riesz space fractional order differential equation, existence and stability as well as convergence, reduced-order extrapolating Crank-Nicolson finite difference scheme, proper orthogonal decomposition.

show abstract

A reduced finite element formulation based on POD method for two-dimensional solute transport problems

Cited by 57 publications

References 19 publications

An indicator‐based problem reduction scheme for coupled reactive transport models

An indicator‐based problem reduction scheme for coupled reactive transport models

A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium

A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Contact Info

Product

Resources

About