Stable Galerkin reduced order models for linearized compressible flow

Barone, Matthew Franklin; Kalashnikova, Irina; Segalman, Daniel J.; Thornquist, Heidi K.

doi:10.1016/j.jcp.2008.11.015

Cited by 169 publications

(204 citation statements)

References 31 publications

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“…Reduced spaces for pressure and velocity fields (denoted respectively Q RB N r and V RB N r ) have the same dimension in the case of physical parametrizations, whereas geometrical parametrizations require modifying the velocity space in order to manage the divergence-free constraint; see Section 3.1. As in the case of problem (1), the functional forms appearing in (14) are obtained by Galerkin projection of the original problem (10) onto the RB space X RB N r = V RB N r × Q RB N r , spanned by the solutions…”

Section: Reduced Basis Construction By Greedy Algorithmsmentioning

confidence: 99%

“…[31,36,37]) Navon et al proposed a dual-weighted POD method, where the weights assigned to each snapshot were derived from an adjoint related to the optimality system of a variational data assimilation problem in meteorology. It is also known that for compressible flows the choice of inner product and weighting of the different flow variables (velocity, pressure, speed of sound) in the snapshot matrix can have a large effect on the stability and accuracy of the ROM [14,35]. Similarly, the H 1 inner product was recommended for the computation of POD modes for compressible Navier-Stokes equations in [66] for the purpose of enhancing stability.…”

Section: "If It Is Not In the Snapshots It Is Not In The Rom"mentioning

confidence: 99%

“…Each iteration entails a large linear system to solve in the case of the truth approximation. Instead, a reduced-order model requires at each iteration of the nonlinear solver the solution of a small linear system, which can be assembled by exploiting the precomputed structures (14).…”

Section: Relevant Computational Issuesmentioning

confidence: 99%

See 2 more Smart Citations

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

161

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Section: Reduced Basis Construction By Greedy Algorithmsmentioning

confidence: 99%

Section: "If It Is Not In the Snapshots It Is Not In The Rom"mentioning

confidence: 99%

See 1 more Smart Citation

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

161

179

View full text Add to dashboard Cite

show abstract

“…Discussed in detail in Lumley [7] and Holmes et al [4], POD is a mathematical procedure that, given an ensemble of data and an inner product, constructs a basis for that ensemble that is optimal in the sense that it describes more energy (on average) of the ensemble in the chosen inner product than any other linear basis of the same dimension M. Stable ROMs based on the Galerkin projection of the fluid equations onto the POD modes requires specific definitions of the inner product [1,8]. ROMs based on the simple L2 inner product can be unstable [1]. For the non-linear Navier-Stokes equations an energy inner product is required for stability [5].…”

Section: Introductionmentioning

confidence: 99%

Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

Fike¹

2013

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The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.

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“…Section 5 details the numerical experiments carried out in order to validate the accuracy of the POD reduced order model and the POD 4-D VAR approach for the various numerical methods enumerated above. For the recent work on POD 4-D VAR, see [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. In particular, we compare ad hoc adaptivity for POD 4-D VAR with trust-region adaptivity in combination with dual-weighted snapshots when full observations are available in our experiment.…”

Section: Introductionmentioning

confidence: 99%