13 research outputs found
Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces
We continue our analysis of the coupling between nonlinear hyperbolic
problems across possibly resonant interfaces. In the first two parts of this
series, we introduced a new framework for coupling problems which is based on
the so-called thin interface model and uses an augmented formulation and an
additional unknown for the interface location; this framework has the advantage
of avoiding any explicit modeling of the interface structure. In the present
paper, we pursue our investigation of the augmented formulation and we
introduce a new coupling framework which is now based on the so-called thick
interface model. For scalar nonlinear hyperbolic equations in one space
variable, we observe that the Cauchy problem is well-posed. Then, our main
achievement in the present paper is the design of a new well-balanced finite
volume scheme which is adapted to the thick interface model, together with a
proof of its convergence toward the unique entropy solution (for a broad class
of nonlinear hyperbolic equations). Due to the presence of a possibly resonant
interface, the standard technique based on a total variation estimate does not
apply, and DiPerna's uniqueness theorem must be used. Following a method
proposed by Coquel and LeFloch, our proof relies on discrete entropy
inequalities for the coupling problem and an estimate of the discrete entropy
dissipation in the proposed scheme.Comment: 21 page
Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flow
Granular materials will segregate by particle size when subjected to shear,
as occurs, for example, in avalanches. The evolution of a bidisperse mixture of
particles can be modeled by a nonlinear first order partial differential
equation, provided the shear (or velocity) is a known function of position.
While avalanche-driven shear is approximately uniform in depth, boundary-driven
shear typically creates a shear band with a nonlinear velocity profile. In this
paper, we measure a velocity profile from experimental data and solve initial
value problems that mimic the segregation observed in the experiment, thereby
verifying the value of the continuum model. To simplify the analysis, we
consider only one-dimensional configurations, in which a layer of small
particles is placed above a layer of large particles within an annular shear
cell and is sheared for arbitrarily long times. We fit the measured velocity
profile to both an exponential function of depth and a piecewise linear
function which separates the shear band from the rest of the material. Each
solution of the initial value problem is non-standard, involving curved
characteristics in the exponential case, and a material interface with a jump
in characteristic speed in the piecewise linear case
Kinetic Formulation and Uniqueness for Scalar Conservation Laws with Discontinuous Flux
We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux