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SUMMARY:Locally conservative approximation of (conservative) systems writt
en in non conservation form: application to Lagrangian hydrodynamics and m
ultifluid problems
DTSTART;VALUE=DATE-TIME:20161212T163000Z
DTEND;VALUE=DATE-TIME:20161212T170000Z
DTSTAMP;VALUE=DATE-TIME:20211016T182818Z
UID:indico-contribution-572@indico.math.cnrs.fr
DESCRIPTION:Speakers: Rémi Abgrall (Université de Zürich)\nSince the ce
lebrated Lax Wendroff theorem\, it is known that the right way of discreti
sing systems of hyperbolic\nequations written in conservation form is to u
se a flux formulation. However\, in many occasions\, the relevant formu-\n
lation\, from an engineering point of view\, is not to consider this conse
rvative formulation but one non conservative\nform. For example\, with sta
ndard notations\, a one fluid model writes\n[équation : voir résumé PDF
] (1)\nbut the interesting quantities are the mass\, velocity and pressure
\, which evolution is described by:\n[équation : voir résumé PDF] (2)\n
Unfortunately this form is not suitable to approximation. In the case of a
multi-fluid system\, the same problem occurs.\nIn this talk\, we will des
cribe a method to overcome this issue. It does not use any flux formulatio
n per se\, but\ncan be shown to provide the right solutions. In order to i
llustrate the method\, we will consider several examples in\nEulerian and
Lagrangian hydrodynamics\nWe will first start from the Residual Distributi
on (RD) (re-)interpretation of the Dobrev et al. scheme [1] for\nthe numer
ical solution of the Euler equations in Lagrangian form. The first ingredi
ent of the original scheme is\nthe staggered grid formulation which uses c
ontinuous node-based finite element approximations for the kinematic\nvari
ables and cell-centered discontinuous finite elements for the thermodynami
c parameters. The second ingredient\nof the Dobrev et al. scheme is an art
ificial viscosity technique applied in order to make possible the computat
ion of\nstrong discontinuities. Using a reformulation in term of RD scheme
\, we can show that the scheme is indeed locally\nconservative while the f
ormulation is stricto sensu non conservative. Using this\, we can generali
se the construction\nand develop locally conservative artificial viscosity
free schemes. To demonstrate the robustness of the proposed RD\nscheme\,
we solve several one-dimensional shock tube problems from rather mild to v
ery strong ones: we go from the\nclassical Sod problem\, to TNT explosions
(with JWL EOS) via the Collela-Woodward blast wave problem.\nIn a second
part\, we show how to extend this method to the Eulerian framework and giv
e applications on single\nfluid and multiphase problems via the five equat
ion model.\nReferences\n[1] V. Dobrev\, T. Kolev\, and R. Rieben. High ord
er curvilinear finite element methods for Lagrangian hydrodynamics.\nSIAM
J. Sci. Comput\, 34:B606–B641\, 2012.\n\nhttps://indico.math.cnrs.fr/eve
nt/1577/contributions/572/
LOCATION:Institut Henri Poincaré Amphi Hermite
URL:https://indico.math.cnrs.fr/event/1577/contributions/572/
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