4 research outputs found
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
Entropy conditions for scalar conservation laws with discontinuous flux revisited
We propose new entropy admissibility conditions for multidimensional
hyperbolic scalar conservation laws with discontinuous flux which generalize
one-dimensional Karlsen-Risebro-Towers entropy conditions.
These new conditions are designed, in particular, in order to characterize
the limit of vanishing viscosity approximations.
On the one hand, they comply quite naturally with a certain class of physical
and numerical modeling assumptions; on the other hand, their mathematical
assessment turns out to be intricate. \smallskip The generalization we propose
is not only with respect to the space dimension, but mainly in the sense that
the "crossing condition" of [K.H. Karlsen, N.H. Risebro, J. Towers,
Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness
with the new definition. We prove uniqueness of solutions and give tools to
justify their existence via the vanishing viscosity method, for the
multi-dimensional spatially inhomogeneous case with a finite number of
Lipschitz regular hypersurfaces of discontinuity for the flux function.Comment: multidimensional case is included and mistakes are correcte
CONVERGENCE OF FINITE VOLUME SCHEMES FOR TRIANGULAR SYSTEMS OF CONSERVATION LAWS
We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type for these systems that exploit the triangular structure. We prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero. Some numerical examples are presented, one of which is related to flows in porous media