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Speaker
Rémi Abgrall, Professor Institut für Mathematik & Computational Science Universität Zürich
Title
Two cases of schemes that use a non conservative version of a conservative system: the Euler equations.
Abstract
In this talk, I will consider two cases of schemes that use a non conservative version of a conservative system, namely the Euler equations for compressible fluid dynamics. The first one uses a staggered representation with a formulation in primitive variables (variables are density, velocity and internal energy, the thermodynamical variables are stored in elements, and the kinetic variables are at the vertices, so that the thermodynamics is locally discontinuous while the kinematics are globally continuous), and I will show a systematic way to recover local conservation, with some numerical examples to show that it works.
The second example is inspired by the Active Flux formulation by Roe and co-authors, but with some important changes. In one dimension, the average values of the conservative variables are updated in the cells, but on the boundary of the cells, we update directly a non conservative system, for example in the primitive variables, or in the variables entropy, velocity, pressure. We also show on numerical examples that it works well.
I addition, in both cases, I show, under standard assumptions, a Lax Wendroff like theorem, though the formulations are a bit non standard. In my opinion, this opens new doors to the numerical discretization of hyperbolic problems. The first part is a joint work with K. Ivanona, and I have beneficiated of discussions with W. Barsukow (CNRS, Bordeaux, France).