A Galerkin/POD Reduced-Order Model from Eigenfunctions of Non-Converged Time Evolution Solutions in a Convection Problem

Cortés, Jesús; Herrero, H.; Plá, F.

doi:10.3390/math10060905

Cited by 5 publications

(2 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The second step: SVD decomposition of sample matrix, through SVD decomposition technology (see, e.g., [26][27][28]), the sample matrix can be expressed as:…”

Section: Galerkin Spectral Analysis Of Two-dimensional Transient Nonl...mentioning

confidence: 99%

Numerical Analysis of Transient State Heat Transfer by Spectral Method Based on POD Reduced-Order Extrapolation Algorithm

Wang

2023

Applied Sciences

View full text Add to dashboard Cite

In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, an iterative scheme of Proper Orthogonal Decomposition (for short, POD) dimension reduction based on the classical central difference Galerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. The POD dimension reduction spectral method model is constructed by taking the calculation results of classical central difference Galerkin spectral method as sample data. The numerical algorithm characteristics of flow and heat transfer are studied by using a partial differential equation as a mathematical model, and the error estimation is given. Finally, different time intervals are used as parameters to simulate experiments. The results show that the POD method is applicable to transient nonlinear heat conduction problems, and the maximum average relative error of the reconstructed temperature field is 0.89675%. Moreover, the POD method not only has a high calculation accuracy, but also has an average calculation speed as high as 310.25 times that of the central difference Galerkin algorithm. It can be seen that under the condition that the error between the solution of POD dimension reduction extrapolation algorithm and the solution of classical central difference Galerkin spectrum method is small enough, the POD method can greatly reduce the calculation amount, shorten the running time, and ensure a high accuracy of the calculation results, thus verifying the effectiveness and feasibility of the algorithm.

show abstract

“…The second step: SVD decomposition of sample matrix, through SVD decomposition technology (see, e.g., [26][27][28]), the sample matrix can be expressed as:…”

Section: Galerkin Spectral Analysis Of Two-dimensional Transient Nonl...mentioning

confidence: 99%

Numerical Analysis of Transient State Heat Transfer by Spectral Method Based on POD Reduced-Order Extrapolation Algorithm

Wang

2023

Applied Sciences

View full text Add to dashboard Cite

show abstract

“…Newton's method considered in this work is the standard method that also provides good results for this type of problems (see Refs. [59][60][61][62][63]).…”

Section: Introductionmentioning

confidence: 99%

2D Newton Schwarz Legendre Collocation Method for a Convection Problem

2022

Self Cite

View full text Add to dashboard Cite

In this work, an alternate Schwarz domain decomposition method is proposed to solve a Rayleigh–Bénard problem. The problem is modeled with the incompressible Navier–Stokes equations coupled with a heat equation in a rectangular domain. The Boussinesq approximation is considered. The nonlinearity is solved with Newton’s method. Each iteration of Newton’s method is discretized with an alternating Schwarz scheme, and each Schwarz problem is solved with a Legendre collocation method. The original domain is divided into several subdomains in both directions of the plane. Legendre collocation meshes are coarse, so the problem in each subdomain is well conditioned, and the size of the total mesh can grow by increasing the number of subdomains. In this way, the ill conditioning of Legendre collocation is overcome. The present work achieves an efficient alternating Schwarz algorithm such that the number of subdomains can be increased indefinitely in both directions of the plane. The method has been validated with a benchmark with numerical solutions obtained with other methods and with real experiments. Thanks to this domain decomposition method, the aspect ratio and Rayleigh number can be increased considerably by adding subdomains. Rayleigh values near to the turbulent regime can be reached. Namely, the great advantage of this method is that we obtain solutions close to turbulence, or in domains with large aspect ratios, by solving systems of linear equations with well-conditioned matrices of maximum size one thousand. This is an advantage over other methods that require solving systems with huge matrices of the order of several million, usually with conditioning problems. The computational cost is comparable to other methods, and the code is parallelizable.

show abstract