9,010 research outputs found
Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension
We prove weak-strong uniqueness results for the compressible Navier-Stokes
system with degenerate viscosity coefficient and with vacuum in one dimension.
In other words, we give conditions on the weak solution constructed in
\cite{Jiu} so that it is unique. The novelty consists in dealing with initial
density which contains vacuum. To do this we use the notion of
relative entropy developed recently by Germain, Feireisl et al and Mellet and
Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the
compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce
a new effective velocity which makes the system parabolic on the density and
hyperbolic on this velocity).Comment: arXiv admin note: text overlap with arXiv:1411.550
A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations
In this paper, a compact third-order gas-kinetic scheme is proposed for the
compressible Euler and Navier-Stokes equations. The main reason for the
feasibility to develop such a high-order scheme with compact stencil, which
involves only neighboring cells, is due to the use of a high-order gas
evolution model. Besides the evaluation of the time-dependent flux function
across a cell interface, the high-order gas evolution model also provides an
accurate time-dependent solution of the flow variables at a cell interface.
Therefore, the current scheme not only updates the cell averaged conservative
flow variables inside each control volume, but also tracks the flow variables
at the cell interface at the next time level. As a result, with both cell
averaged and cell interface values the high-order reconstruction in the current
scheme can be done compactly. Different from using a weak formulation for
high-order accuracy in the Discontinuous Galerkin (DG) method, the current
scheme is based on the strong solution, where the flow evolution starting from
a piecewise discontinuous high-order initial data is precisely followed. The
cell interface time-dependent flow variables can be used for the initial data
reconstruction at the beginning of next time step. Even with compact stencil,
the current scheme has third-order accuracy in the smooth flow regions, and has
favorable shock capturing property in the discontinuous regions. Many test
cases are used to validate the current scheme. In comparison with many other
high-order schemes, the current method avoids the use of Gaussian points for
the flux evaluation along the cell interface and the multi-stage Runge-Kutta
time stepping technique.Comment: 27 pages, 38 figure
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