103 research outputs found
Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme
For a model of nonlinear elastodynamics, we construct a finite volume scheme
which is able to capture nonclassical shocks (also called undercompressive
shocks). Those shocks verify an entropy inequality but are not admissible in
the sense of Liu. They verify a kinetic relation which describes the jump, and
keeps an information on the equilibrium between a vanishing dispersion and a
vanishing diffusion. The scheme pre-sented here is by construction exact when
the initial data is an isolated nonclassical shock. In general, it does not
introduce any diffusion near shocks, and hence nonclas-sical solutions are
correctly approximated. The method is fully conservative and does not use any
shock-tracking mesh. This approach is tested and validated on several test
cases. In particular, as the nonclassical shocks are not diffused at all, it is
possible to obtain large time asymptotics
Convergent and conservative schemes for nonclassical solutions based on kinetic relations
We propose a new numerical approach to compute nonclassical solutions to
hyperbolic conservation laws. The class of finite difference schemes presented
here is fully conservative and keep nonclassical shock waves as sharp
interfaces, contrary to standard finite difference schemes. The main challenge
is to achieve, at the discretization level, a consistency property with respect
to a prescribed kinetic relation. The latter is required for the selection of
physically meaningful nonclassical shocks. Our method is based on a
reconstruction technique performed in each computational cell that may contain
a nonclassical shock. To validate this approach, we establish several
consistency and stability properties, and we perform careful numerical
experiments. The convergence of the algorithm toward the physically meaningful
solutions selected by a kinetic relation is demonstrated numerically for
several test cases, including concave-convex as well as convex-concave
flux-functions.Comment: 31 page
Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient
We are concerned with fully-discrete schemes for the numerical approximation
of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux
function in one-space dimension. More precisely, we show the convergence of
approximate solutions, generated by the scheme corresponding to vanishing
diffusive-dispersive scalar conservation laws with a discontinuous coefficient,
to the corresponding scalar conservation law with discontinuous coefficient.
Finally, the convergence is illustrated by several examples. In particular, it
is delineated that the limiting solutions generated by the scheme need not
coincide, depending on the relation between diffusion and the dispersion
coefficients, with the classical Kruzkov-Oleinik entropy solutions, but contain
nonclassical undercompressive shock waves.Comment: 38 Pages, 6 figure
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