Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Consistent Internal Energy Based Schemes for the Compressible Euler
  Equations   Numerical schemes for the solution of the Euler equations have recently been
developed, which involve the discretisation of the internal energy equation,
with corrective terms to ensure the correct capture of shocks, and, more
generally, the consistency in the Lax-Wendroff sense. These schemes may be
staggered or colocated, using either struc-tured meshes or general simplicial
or tetrahedral/hexahedral meshes. The time discretization is performed by
fractional-step algorithms; these may be either based on semi-implicit pressure
correction techniques or segregated in such a way that only explicit steps are
involved (referred to hereafter as "explicit" variants). In order to ensure the
positivity of the density, the internal energy and the pressure, the discrete
convection operators for the mass and internal energy balance equations are
carefully designed; they use an upwind technique with respect to the material
velocity only. The construction of the fluxes thus does not need any Rie-mann
or approximate Riemann solver, and yields easily implementable algorithms. The
stability is obtained without restriction on the time step for the pressure
correction scheme and under a CFL-like condition for explicit variants:
preservation of the integral of the total energy over the computational domain,
and positivity of the density and the internal energy. The semi-implicit
first-order upwind scheme satisfies a local discrete entropy inequality. If a
MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker
property holds: the entropy inequality is satisfied up to a remainder term
which is shown to tend to zero with the space and time steps, if the discrete
solution is controlled in L $\infty$ and BV norms. The explicit upwind variant
also satisfies such a weaker property, at the price of an estimate for the
velocity which could be derived from the introduction of a new stabilization
term in the momentum balance. Still for the explicit scheme, with the
above-mentioned MUSCL-like scheme, the same result only holds if the ratio of
the time to the space step tends to zero.

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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph