Observers for Compressible Navier--Stokes Equation

Apte, Amit; Auroux, Didier; Ramaswamy, Mythily

doi:10.1137/16m1060601

Cited by 3 publications

(10 citation statements)

References 40 publications

(32 reference statements)

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“…for some constant 𝐶, which follows from the fact that  2 𝜕 2 𝑥 in Fourier space is the multiplier −𝑘 2 (𝑘) 2 and  is a positive operator (see definition in (26) or (28)).…”

Section: Bathymetry Using Surface Datamentioning

confidence: 99%

“…Remark 7. So far we have only discussed reconstruction with relation to the dispersive models of ( 26) and (28). We now briefly consider the hyperbolic model corresponding to the choice 𝜔 2 = 𝑘 2 and  = 1.…”

Section: Bathymetry Using Surface Datamentioning

confidence: 99%

“…We discretized this function on a uniform grid in the 𝑥-variable with 256 points. For fixed 𝑡, using a pseudospectral method, we compute the matrix representation of the operator (𝑞 𝑥 ) 2 𝜕 2 𝑥 ((𝑞 𝑥 )⋅) for both regularized Boussinesq (26) and regularized Boussinesq-Whitham (28). This generates a 256 × 256 real symmetric matrix for either model.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The observer framework is popular in the geophysics community and is routinely used in data assimilation to recover the state of a dynamical system from partial or sparse measurements. [27][28][29][30] We then combine the two inverse problems in Section 5 to design our algorithm for bottom-boundary reconstruction from surface-deviation measurements alone. We do not require the mean depth to be known, and additionally, we are able to reconstruct bottom profiles in deeper water than in Ref.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ocean‐depth measurement using shallow‐water wave models

2021

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In this paper, we consider a problem inspired by the real-world need to identify the topographical features of ocean basins. Specifically, we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free-surface deviation alone, within the context of dispersive shallow-water wave models. The need to consider the shallow-water regime arises from the ill-posed nature of the problem and is motivated by prior work. Assuming only a relatively inaccurate initial guess for the bottom-boundary, we design an algorithm to accurately deduce the fluid velocities and the true bottom-boundary profile. We achieve this by considering two separate inverse problems: one to deduce the bottom-boundary from velocities and the surface deviation, and another to recover the velocities from the surface deviation and an approximate bottom-boundary.The former is a classic inverse problem that requires the inversion of an ill-conditioned matrix, whereas the latter employs the observer framework. Combining the two inverse problems leads to our reconstruction algorithm. We emphasize the role played by model selection and its impact on algorithm design and the accuracy of the reconstruction.

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Section: Bathymetry Using Surface Datamentioning

confidence: 99%

Section: Bathymetry Using Surface Datamentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ocean‐depth measurement using shallow‐water wave models

2021

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“…Here we have used a very simple nudging matrix. Further investigation could be performed to develop more complex observers, as in Auroux & Bonnabel (2011), Apte et al (2018) or Krstic et al (2009), in order to correct fluid height with velocity observations.…”

Section: Data Assimilation At Local Scale To Improve Cfd Simulations mentioning

confidence: 99%

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Defforge

Carissimo

Bocquet

et al. 2018

IJEP

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Atmospheric dispersion modelling requires meteorological inputs over local domains with possibly complex topographies. These local wind fields may be difficult to simulate with CFD models, in particular because of their sensitivity to geometrical features and to model inputs, especially the boundary conditions which are generally provided by larger-scale models or measurements. Using data assimilation, a few measurements inside the domain could add information to the imprecise boundary conditions and thus greatly enhance the precision of the dispersion simulations. Three data assimilation techniques (3DVar, the back and forth nudging algorithm, and the iterative ensemble Kalman smoother) have been adapted to local scale simulations by taking boundary conditions into account instead of initial conditions for which they are usually applied. Their performances have been evaluated at small scales, with a simple representation of the atmosphere into two layers, using 1D solution of the shallow water equations.

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Ocean-depth measurement using shallow-water wave models

Vasan¹,

Manisha²,

Auroux³

2021

Preprint

Self Cite

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In this paper, we consider a problem inspired by the real-world need to identify the topographical features of ocean basins. Specifically we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free-surface deviation alone, within the context of dispersive shallow-water wave models. The need to consider the shallow-water regime arises from the ill-posed nature of the problem and is motivated by prior work. We design an algorithm using which, both fluid velocities and the bottom-boundary profile, may be accurately recovered assuming an a priori relatively inaccurate guess for the bottom boundary. We achieve this by considering two separate inverse problems: one to deduce the bottom-boundary from velocities and the surface deviation, and another to recover the velocities from the surface deviation and an approximate bottom-boundary. The former is a classic inverse problem that requires the inversion of an ill-conditioned matrix while the latter employs the observer framework. Combining the two inverse problems leads to our reconstruction algorithm. We emphasise the role played by model selection and its impact on algorithm design and the accuracy of the reconstruction.

show abstract

Observers for Compressible Navier--Stokes Equation

Cited by 3 publications

References 40 publications

Ocean‐depth measurement using shallow‐water wave models

Ocean‐depth measurement using shallow‐water wave models

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Ocean-depth measurement using shallow-water wave models

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