%0 Unpublished work
%T Analysis for the fast vector penalty-projection solver of incompressible multiphase Navier-Stokes/Brinkman problems
%+ Institut de Mathématiques de Marseille (I2M)
%+ Institut de Mécanique et d'Ingénierie de Bordeaux (I2M)
%+ Institut de Mathématiques de Bordeaux (IMB)
%A Angot, Philippe
%A Caltagirone, Jean-Paul
%A Fabrie, Pierre
%8 2015-09-05
%D 2015
%K Vector penalty-projection method
%K divergence-free penalty-projection
%K penalty method
%K splitting prediction-correction scheme
%K fast Helmholtz-Hodge decompositions
%K Navier-Stokes/Brinkman equations
%K stability analysis
%K incompressible homogeneous flows
%K dilatable flows
%K low Mach number flows
%K incompressible non-homogeneous or multiphase flows
%Z MSC 2010: 35Q30; 35Q35; 65M12; 65M85; 65N12; 65N85; 74F10; 76D05; 76D45; 76M25; 76R10; 76S05; 76T10
%Z Mathematics [math]/Analysis of PDEs [math.AP]
%Z Mathematics [math]/Classical Analysis and ODEs [math.CA]
%Z Mathematics [math]/Numerical Analysis [math.NA]
%Z Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]
%Z Engineering Sciences [physics]
%Z Mathematics [math]
%Z Physics [physics]
%Z Physics [physics]/Mechanics [physics]
%Z Engineering Sciences [physics]/Reactive fluid environmentPreprints, Working Papers, ...
%X 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
%G English
%2 https://hal.science/hal-01194345/document
%2 https://hal.science/hal-01194345/file/Numer_Math_ACF2015.pdf
%L hal-01194345
%U https://hal.science/hal-01194345
%~ CNRS
%~ UNIV-AMU
%~ ENSAM
%~ INRA
%~ IMB
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ TDS-MACS
%~ AGREENIUM
%~ I2M-BX
%~ INRAE
%~ HESAM
%~ HESAM-ENSAM
%~ IRENAV
%~ LAMPA
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%~ LABOMAP
%~ LISPEN
%~ MSMP