Analysis for the fast vector penalty-projection solver of incompressible multiphase Navier-Stokes/Brinkman problems
| hal.structure.identifier | Institut de Mathématiques de Marseille [I2M] | |
| dc.contributor.author | ANGOT, Philippe | |
| dc.contributor.author | CALTAGIRONE, Jean-Paul | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | FABRIE, Pierre | |
| dc.date.accessioned | 2021-05-14T09:56:16Z | |
| dc.date.available | 2021-05-14T09:56:16Z | |
| dc.date.created | 2015-09-05 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/77763 | |
| dc.description.abstractEn | We detail and theoretically analyse the so-called fast vector (or velocity) penalty-projection methods (VPP ε) of which the main ideas and features are briefly introduced in [8,9,10]. This family of numerical schemes proves to efficiently compute the solution of unsteady Navier-Stokes/Brinkman problems governing incompressible or low Mach multi-phase viscous flows with variable mass density and/or viscosity or anisotropic permeability. In this paper, we describe in detail the connections and essential differences with usual methods to solve the Navier-Stokes equations. The key idea of the basic (VPP ε) method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This is obtained by proposing new Helmholtz-Hodge decomposition solutions of L 2-vector fields in bounded domains to get fast methods with suitable adapted right-hand sides; see [11]. This procedure only requires a few iterations of preconditioned conjugate gradients whatever the spatial mesh step. Then, the splitting (VPP ε) method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as O(ε δt), δt being the time step, with a penalty parameter ε as small as desired until the machine precision, e.g. ε = 10 −14 , whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed velocity correction step typically requires only one or two iterations of a suitable pre-conditioned Krylov solver whatever the spatial mesh step [10]. Moreover, the robustness of our method is not sensitive to large mass density ratios since the velocity penalty-projection step does not include any spatial derivative of the density. 2 In the present work, we also prove the theoretical foundations as well as global sol-vability and optimal unconditional stability results of the (VPP ε) method for Navier-Stokes problems in the case of homogeneous flows, which are the main new results. Keywords Vector penalty-projection method · divergence-free penalty-projection · penalty method · splitting prediction-correction scheme · fast Helmholtz-Hodge decompositions · Navier-Stokes/Brinkman equations · stability analysis · incompressible homogeneous flows · dilatable flows · low Mach number flows · incompressible non-homogeneous or multiphase flows | |
| dc.language.iso | en | |
| dc.subject.en | Vector penalty-projection method | |
| dc.subject.en | divergence-free penalty-projection | |
| dc.subject.en | penalty method | |
| dc.subject.en | splitting prediction-correction scheme | |
| dc.subject.en | fast Helmholtz-Hodge decompositions | |
| dc.subject.en | Navier-Stokes/Brinkman equations | |
| dc.subject.en | stability analysis | |
| dc.subject.en | incompressible homogeneous flows | |
| dc.subject.en | dilatable flows | |
| dc.subject.en | low Mach number flows | |
| dc.subject.en | incompressible non-homogeneous or multiphase flows | |
| dc.title.en | Analysis for the fast vector penalty-projection solver of incompressible multiphase Navier-Stokes/Brinkman problems | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.subject.hal | Mathématiques [math]/Analyse classique [math.CA] | |
| dc.subject.hal | Mathématiques [math]/Analyse numérique [math.NA] | |
| dc.subject.hal | Physique [physics]/Mécanique [physics]/Mécanique des fluides [physics.class-ph] | |
| dc.subject.hal | Sciences de l'ingénieur [physics] | |
| dc.subject.hal | Mathématiques [math] | |
| dc.subject.hal | Physique [physics] | |
| dc.subject.hal | Physique [physics]/Mécanique [physics] | |
| dc.subject.hal | Sciences de l'ingénieur [physics]/Milieux fluides et réactifs | |
| bordeaux.hal.laboratories | Institut de Mécanique et d’Ingénierie de Bordeaux (I2M) - UMR 5295 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.institution | INRAE | |
| bordeaux.institution | Arts et Métiers | |
| hal.identifier | hal-01194345 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01194345v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=ANGOT,%20Philippe&CALTAGIRONE,%20Jean-Paul&FABRIE,%20Pierre&rft.genre=preprint |
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