2,222 research outputs found
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes
This series of papers is devoted to the formulation and the approximation of
coupling problems for nonlinear hyperbolic equations. The coupling across an
interface in the physical space is formulated in term of an augmented system of
partial differential equations. In an earlier work, this strategy allowed us to
develop a regularization method based on a thick interface model in one space
variable. In the present paper, we significantly extend this framework and, in
addition, encompass equations in several space variables. This new formulation
includes the coupling of several distinct conservation laws and allows for a
possible covering in space. Our main contributions are, on one hand, the design
and analysis of a well-balanced finite volume method on general triangulations
and, on the other hand, a proof of convergence of this method toward entropy
solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a
single conservation law without coupling). The core of our analysis is, first,
the derivation of entropy inequalities as well as a discrete entropy
dissipation estimate and, second, a proof of convergence toward the entropy
solution of the coupling problem.Comment: 37 page
Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution
We consider an immiscible two-phase flow in a heterogeneous one-dimensional
porous medium. We suppose particularly that the capillary pressure field is
discontinuous with respect to the space variable. The dependence of the
capillary pressure with respect to the oil saturation is supposed to be weak,
at least for saturations which are not too close to 0 or 1. We study the
asymptotic behavior when the capillary pressure tends to a function which does
not depend on the saturation. In this paper, we show that if the capillary
forces at the spacial discontinuities are oriented in the same direction that
the gravity forces, or if the two phases move in the same direction, then the
saturation profile with capillary diffusion converges toward the unique optimal
entropy solution to the hyperbolic scalar conservation law with discontinuous
flux functions
L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method
We consider scalar nonviscous conservation laws with strictly convex flux in
one spatial dimension, and we investigate the behavior of bounded L^2
perturbations of shock wave solutions to the Riemann problem using the relative
entropy method. We show that up to a time-dependent translation of the shock,
the L^2 norm of a perturbed solution relative to the shock wave is bounded
above by the L^2 norm of the initial perturbation.Comment: 17 page
On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws
We prove convergence of a class of space-time discontinuous Galerkin schemes
for scalar hyperbolic conservation laws. Convergence to the unique entropy
solution is shown for all orders of polynomial approximation, provided strictly
monotone flux functions and a suitable shock-capturing operator are used. The
main improvement, compared to previously published results of similar scope, is
that no streamline-diffusion stabilization is used. This is the way
discontinuous Galerkin schemes were originally proposed, and are most often
used in practice
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