A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids

Vidović, D.; Segal, A.; Wesseling, Pieter

doi:10.1016/j.jcp.2006.01.031

Cited by 32 publications

(24 citation statements)

References 21 publications

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“…In short, in the collocated scheme, the faces mass flow have to be computed using Eq. (22). Analogously, in the staggered scheme, the cell-centred velocities have to be reconstructed from the face-centred ones using either STAGG1 or STAGG2 reconstruction method.…”

Section: Predictor -First Step L D 1/mentioning

confidence: 99%

“…(22). Analogously, in staggered schemes, the cell-centred velocities u nC1 i;c have to be reconstructed from the face-centred mass fluxes using either STAGG1 or STAGG2 reconstruction method.…”

Section: Corrector -Second Step L D 2/mentioning

confidence: 99%

“…However, construction of such meshes is not straightforward. Extension to arbitrary unstructured meshes using staggered approaches but for incompressible flows has also been considered [21,22]. Concerning collocated schemes, Lien [9] proposed a segregated algorithm for all-speed flows, Mahesh et al initially developed an scheme for incompressible flows on complex geometries [23] and later extended it to the variable density case [24], and Shunn et al [10] using a similar approach, proposed a verification process for low Mach algorithms where unstructured meshes were used.…”

Section: Introductionmentioning

confidence: 99%

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Numerical analysis of conservative unstructured discretisations for low Mach flows

Ventosa-Molina

Chiva

Lehmkuhl

et al. 2016

Numerical Methods in Fluids

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Summary Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Predictor -First Step L D 1/mentioning

confidence: 99%

Section: Corrector -Second Step L D 2/mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical analysis of conservative unstructured discretisations for low Mach flows

Ventosa-Molina

Chiva

Lehmkuhl

et al. 2016

Numerical Methods in Fluids

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“…The development of pressure correction techniques for compressible Navier-Stokes equations may be traced back to the seminal work of Harlow and Amsden [20,21] in the late sixties, who developed an iterative algorithm (the so-called ICE method) including an elliptic corrector step for the pressure. Later on, pressure correction equations appeared in numerical schemes proposed by several researchers, essentially in the finite-volume framework, using either a collocated [10,23,26,30,33,34] or a staggered arrangement [2,4,7,22,24,25,37,38,[40][41][42] of unknowns; in the first case, some corrective actions are to be foreseen to avoid the usual odd-even decoupling of the pressure in the low Mach number regime. Some of these algorithms are essentially implicit, since the final stage of a time step involves the unknown at the end-of-step time level; the end-of-step solution is then obtained by SIMPLE-like iterative processes [10,23,25,26,30,34,39].…”

Section: Introductionmentioning

confidence: 99%

“…Some of these algorithms are essentially implicit, since the final stage of a time step involves the unknown at the end-of-step time level; the end-of-step solution is then obtained by SIMPLE-like iterative processes [10,23,25,26,30,34,39]. The other schemes [2,7,22,24,33,37,38,40,42,43] are predictor-corrector methods, where basically two steps are performed sequentially: first a semi-explicit decoupled prediction of the momentum or velocity (and possibly energy, for non-barotropic flows) and, second, a correction step where the end-of step pressure is evaluated and the momentum and velocity are corrected, as in projection methods for incompressible flows (see [5,36] for the original papers, [29] for a comprehensive introduction and [19] for a review of most variants). The Characteristic-Based Split (CBS) scheme (see [31] for a recent review or [44] for the seminal paper), developed in the finite-element context, belongs to this latter class of methods.…”

Section: Introductionmentioning

confidence: 99%

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

Gallouët¹,

Gastaldo²,

Herbin³

et al. 2008

ESAIM: M2AN

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Abstract. We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2 -stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.Mathematics Subject Classification. 35Q30, 65N12, 65N30, 76M25.

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A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

Yoshioka

Unami

Fujihara

2014

Numerical Methods in Fluids

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SUMMARYAnalysis of surface water flows is of central importance in understanding and predicting a wide range of water engineering issues. Dynamics of surface water is reasonably well described using the shallow water equations (SWEs) with the hydrostatic pressure assumption. The SWEs are nonlinear hyperbolic partial differential equations that are in general required to be solved numerically. Application of a simple and efficient numerical model is desirable for solving the SWEs in practical problems. This study develops a new numerical model of the depth‐averaged horizontally 2D SWEs referred to as 2D finite element/volume method (2D FEVM) model. The continuity equation is solved with the conforming, standard Galerkin FEM scheme and momentum equations with an upwind, cell‐centered finite volume method scheme, utilizing the water surface elevation and the line discharges as unknowns aligned in a staggered manner. The 2D FEVM model relies on neither Riemann solvers nor high‐resolution algorithms in order to serve as a simple numerical model. Water at a rest state is exactly preserved in the model. A fully explicit temporal integration is achieved in the model using an efficient approximate matrix inversion method. A series of test problems, containing three benchmark problems and three experiments of transcritical flows, are carried out to assess accuracy and versatility of the model. Copyright © 2014 John Wiley & Sons, Ltd.

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A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids

Cited by 32 publications

References 21 publications

Numerical analysis of conservative unstructured discretisations for low Mach flows

Numerical analysis of conservative unstructured discretisations for low Mach flows

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

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