64 research outputs found
An Integro-Differential Conservation Law arising in a Model of Granular Flow
We study a scalar integro-differential conservation law. The equation was
first derived in [2] as the slow erosion limit of granular flow. Considering a
set of more general erosion functions, we study the initial boundary value
problem for which one can not adapt the standard theory of conservation laws.
We construct approximate solutions with a fractional step method, by
recomputing the integral term at each time step. A-priori L^\infty bounds and
BV estimates yield convergence and global existence of BV solutions.
Furthermore, we present a well-posedness analysis, showing that the solutions
are stable in L^1 with respect to the initial data
Some new well-posedness results for continuity and transport equations, and applications to the chromatography system
We obtain various new well-posedness results for continuity and transport
equations, among them an existence and uniqueness theorem (in the class of
strongly continuous solutions) in the case of nearly incompressible vector
fields, possibly having a blow-up of the BV norm at the initial time. We apply
these results (valid in any space dimension) to the k x k chromatography system
of conservation laws and to the k x k Keyfitz and Kranzer system, both in one
space dimension.Comment: 33 pages, minor change
A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications
International audienceWe state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution a entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (Comparison Theorem). We next study a BGK-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the Comparison Theorem
On the concentration of entropy for scalar conservation laws
We prove that the entropy for an -solution to a scalar conservation laws with continuous initial data is concentrated on a countably -rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux
A fully adaptive finite volume multiresolution scheme for one-dimensional
strongly degenerate parabolic equations with discontinuous flux is presented.
The numerical scheme is based on a finite volume discretization using the
Engquist--Osher approximation for the flux and explicit time--stepping. An
adaptivemultiresolution scheme with cell averages is then used to speed up CPU
time and meet memory requirements. A particular feature of our scheme is the
storage of the multiresolution representation of the solution in a dynamic
graded tree, for the sake of data compression and to facilitate navigation.
Applications to traffic flow with driver reaction and a clarifier--thickener
model illustrate the efficiency of this method
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas
Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations
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