Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations

Ueckermann, Mattheus P.; Lermusiaux, Pierre F. J.

doi:10.1016/j.jcp.2015.11.028

Cited by 36 publications

(17 citation statements)

References 56 publications

(160 reference statements)

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“…The discrete local mass balance (66b) is a straightforward consequence of (52b) expanding D k T according to its definition (18). As a matter of fact, accounting for Remark 12, we can take as a test function q h such that, for a selected mesh element T ∈ T h , q T spans P k (T ) while q T = 0 for all T ∈ T h \ {T }.…”

Section: Flux Formulationmentioning

confidence: 99%

“…On the other hand, recalling the definition (36) of b h and expanding, for all T ∈ T h , D k T v T according to (18) with = k and q = π k T q |T , we have that…”

Section: Flux Formulationmentioning

confidence: 99%

“…A second family of discretisation methods worth mentioning here are Hybridisable Discontinuous Galerkin (HDG) methods, which can be regarded as an evolution of DG methods where both face and element unknowns are present. Their application to the incompressible Navier-Stokes equations has been considered, e.g., in [15,16,17,18,19]; see also [20] (and, in particular, Remark 5 therein) for a comparison with HHO methods. More recent polyhedral technologies have also been applied to the discretisation of the incompressible Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

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A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

Botti

Pietro

Droniou

2019

Journal of Computational Physics

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In this work we propose a novel Hybrid High-Order method for the incompressible Navier-Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in h k+1 (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two-and three-dimensional lid-driven cavity problems).

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Section: Flux Formulationmentioning

confidence: 99%

“…On the other hand, recalling the definition (36) of b h and expanding, for all T ∈ T h , D k T v T according to (18) with = k and q = π k T q |T , we have that…”

Section: Flux Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

Botti

Pietro

Droniou

2019

Journal of Computational Physics

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“…The HDG method for the incompressible Euler and Navier-Stokes equations was introduced in [81,87], and further developed in [58,68,80,83,98,111]. An analysis of the HDG method for the steady-state incompressible Navier-Stokes equations is presented in [9].…”

Section: Bibliography Notesmentioning

confidence: 99%

Hybridized Discontinuous Galerkin Methods for Wave Propagation

et al. 2018

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We present the recent development of hybridizable and embedded discontinuous Galerkin (DG) methods for wave propagation problems in fluids, solids, and electromagnetism. In each of these areas, we describe the methods, discuss their main features, display numerical results to illustrate their performance, and conclude with bibliography notes. The main ingredients in devising these DG methods are (i) a local Galerkin projection of the underlying partial differential equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; (ii) a judicious choice of the numerical flux to provide stability and consistency; and (iii) a global jump condition that enforces the continuity of the numerical flux to obtain a global system in terms of the numerical trace. These DG methods are termed hybridized DG methods, because they are amenable to hybridization (static condensation) and hence to more efficient implementations. They share many common advantages of DG methods and possess some unique features that make them well-suited to wave propagation problems. *

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“…Its modeling capabilities include implicit two-way nesting for multiscale hydrostatic primitive equation (PE) dynamics with a nonlinear free-surface [97] and a high-order finite element code on unstructured grids for non-hydrostatic processes also with a nonlinear free-surface [98][99][100]. Other MSEAS subsystems include: initialization schemes [101], nested data-assimilative tidal prediction and inversion [102]; fast-marching coastal objective analysis [103]; stochastic subgrid-scale models (e.g., [104,105]); generalized adaptable biogeochemical modeling systems; Lagrangian Coherent Structures; non-Gaussian data assimilation and adaptive sampling [106][107][108]; dynamically-orthogonal equations for uncertainty predictions [109][110][111]; and machine learning of model formulations [112].…”

Section: Mseas Modeling System E Multidisciplinary Simulation Estimmentioning

confidence: 99%

Autonomous and adaptive oceanographic feature tracking on board autonomous underwater vehicles

Petillo

Schmidt

Lermusiaux

et al. 2015

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Abstracte capabilities of autonomous underwater vehicles (AUVs) and their ability to perform tasks both autonomously and adaptively are rapidly improving, and the desire to quickly and efficiently sample the ocean environment as Earth's climate changes and natural disasters occur has increased significantly in the last decade. As such, this thesis proposes to develop a method for single and multiple AUVs to collaborate autonomously underwater while autonomously adapting their motion to changes in their local environments, allowing them to sample and track various features of interest with greater efficiency and synopticity than previously possible with preplanned AUV or ship-based surveys. is concept is demonstrated to work in field testing on multiple occasions: with a single AUV autonomously and adaptively tracking the depth range of a thermocline or acousticline, and with two AUVs coordinating their motion to collect a data set in which internal waves could be detected. is research is then taken to the next level by exploring the problem of adaptively and autonomously tracking spatiotemporally dynamic underwater fronts and plumes using individual and autonomously collaborating AUVs. Stephanie entered the Massachusetts Institute of Technology/Woods Hole Oceanographic Institution Joint Graduate Program following college to study Oceanographic Engineering. Her Doctoral research, which comprises this thesis, has focused on using autonomous underwater vehicles (AUVs) to perform autonomous and environmentally adaptive sampling of the ocean environment, focusing on underwater feature detection and tracking for more efficient and synoptic data collection with AUVs. Stephanie has enjoyed being on and near the ocean, working with oceanographic vehicles and instrumentation, and seeing her complex underwater vehicle missions work out in the water, and she has had the great opportunity of participating in over a dozen research cruises since she first started working with underwater vehicles in college. roughout graduate school, Stephanie has continued her many hobbies and activities, adding hiking, farming, sailing, and woodworking to the list. I would also like to thank all the members of the LAMSS lab past and present: ank you Toby for fixing nearly any software problem we threw at you on cruise and in the office, and for completing our A-team for every cruise we were on together. ank you also for being my partner in crime on cruise, on travel, and in life. I could not have completed this research nearly as efficiently without you there to devise all sorts of work-arounds when I wanted extra features to make my virtual experiments easier to run. ank you to Erin and Sheida for being both great friends and lab mates and for teaching me that it is okay to not be the only woman in an engineering lab! anks Stephanie for always being excited about my research and hosting craft nights. om, thanks for being such a quick learner on cruise and generally a fun person to hang out with. AcknowledgmentsYou will be a good rep...

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Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations

Cited by 36 publications

References 56 publications

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

Hybridized Discontinuous Galerkin Methods for Wave Propagation

Autonomous and adaptive oceanographic feature tracking on board autonomous underwater vehicles

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