Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

Vila-Pérez, Jordi; Giacomini, Matteo; Sevilla, Rubén; Huerta, Antonio

doi:10.1007/s11831-020-09508-z

Cited by 37 publications

(42 citation statements)

References 130 publications

(251 reference statements)

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“…The HDG scheme has been applied to the discretization of the compressible Navier-Stokes system (La Spina et al, 2020;Peraire et al, 2010;Woopen et al, 2014) and the weakly compressible Navier-Stokes system (Vila-Pérez et al, 2021). However, we are concerned with the weakly compressible Stokes system.…”

Section: Hdg Methodsmentioning

confidence: 99%

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

Sime

Wilson

Keken

2022

Geochem Geophys Geosyst

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Plate tectonics causes continuous differentiation of the Earth's mantle. At mid-oceanic ridges decompression melting forms a basaltic crust on top of a depleted peridotite. Melting at subduction zones helps generate the continental crust and provides further chemical modifications to the subducting crust and mantle. Subduction introduces the newly formed heterogeneity to the deep mantle where it remixes with older subducted materials as well as the likely remnants of early Earth formation, differentiation, and magma ocean solidification (Elkins-Tanton, 2008;Labrosse et al., 2007). Large scale mantle convection associated with plate tectonics also remixes past and newly generated heterogeneities leading to a "marble cake" mantle (Allègre & Turcotte, 1986;van Keken et al., 2014) with its complicated geochemical history expressed in the broad heterogeneity of mid-oceanic ridge and ocean island basalts (Hofmann, 2014;van Keken et al., 2002;Zindler & Hart, 1986), with indications of preservation of heterogeneity caused by very early differentiation of the Earth's mantle (Boyet & Carlson, 2005) even in modern basalts (Horan et al., 2018). The long-term recycling of oceanic crust mixed in with the depleted harzburgite-derived component and ambient mantle is an attractive explanation (Christensen & Hofmann, 1994;Jones et al., 2020) for the bulk of the seismologically observed Large Low Shear Velocity Provinces (LLSVPs: Garnero & McNamara, 2008) at the base of the mantle. However, it is as yet not clear how significant a contribution any primordial heterogeneity has to these LLSVPs (Ballmer et al., 2016;Li et al., 2014) and whether other, potentially more exotic, processes are at play to maintain chemical heterogeneity (Ballmer et al., 2017;Kellogg et al., 1999).To aid in the understanding of the thermal and chemical evolution of the Earth's mantle we can test hypotheses using predictive models of mantle convection driven by thermal and chemical buoyancy forces. This requires numerical methods to reliably model the evolution of temperature and chemistry that are based on the numerical solution of the governing equations following from conservation of momentum, mass, energy, and chemical species. Significant work has relied on the incompressible Boussinesq approximation (e.g.

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Section: Hdg Methodsmentioning

confidence: 99%

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

Sime

Wilson

Keken

2022

Geochem Geophys Geosyst

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“…In addition, an entropy-stable space-time discretisation was proposed for the compressible Navier-Stokes equations using an HDG approach in space and a discontinuous approximation in time [266]. More recently, special attention was dedicated to the development of positivity-preserving Riemann solvers in the context of hybridised DG methods [263]. For a complete review on HDG methods for compressible flows, interested readers are referred to [263], whereas the application to gas kinetics modelled by means of the linearised Bhatnagar-Gross-Krook equation is discussed in [254].…”

Section: Compressible Flows and Gas Kinetics Equationsmentioning

confidence: 99%

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

Giacomini

Sevilla

Huerta

2020

Arch Computat Methods Eng

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This paper presents , an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in . Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator is provided to facilitate its application to practical engineering problems.

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“…Indeed, as we shall demonstrate by means of numerical test cases, the formulation is able to cope with the incompressible Euler equations. Inspired by DG methods based on Godunov fluxes, see e.g., [11,12,42,53] and following the ideas proposed by [61] we employ an Harten, Lax and van Leer (HLL) approximated Riemann solver for designing pressure-velocity coupling and convective term treatment. The scheme relies on polynomials of degree 𝑘 for both elemental and skeletal unknowns resulting in convergence rates of order 𝑘 + 1 for both the velocity and pressure error in 𝐿 2 norm.…”

Section: Introductionmentioning

confidence: 99%

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

Botti¹,

Massa²

2021

Preprint

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We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL) approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on ℎ-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.

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Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

Abstract: This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers

Cited by 37 publications

References 130 publications

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

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