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Hamiltonianpreserving schemes for the Liouville equation with discontinuous potentials
"... When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, We introduce two classes o ..."
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When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, We introduce two classes of Hamiltonianpreserving schemes for such problems. By using the constant Hamiltonian across the potential barrier, we introduced a selection criterion for a unique, physically relavant solution to the underlying linear hyperbolic equation with singular coefficients. These scheme have a hyperbolic CFL condition, which is a significant improvement over a conventional discretization. We also establish the positivity, and stability in both l 1 and l ∞ norms, of these discretizations, and conducted numerical experiments to study the numerical accuracy. This work is motivated by the wellbalanced kinetic schemes by Perthame and Simeoni for the shallow water equations with a discontinuous bottom topography, and has applications to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrödinger equation with a discontinuous potential, among other applications.
Quasilinear anisotropic degenerate parabolic equations with timespace dependent diffusion coefficients
 Commun. Pure Appl. Anal
, 2005
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Uniqueness of L∞ solutions for a class of conormal BV vector fields
 Contemp. Math
"... A bstract. Let X be a bounded vector field with bounded divergence defined in an open set Ω of Rd, transverse to a hypersurface S. Let Ω0 be an open subset of Ω such that the Hausdorff measure Hd−1(Ω\Ω0) = 0. We suppose that the vector field X belongs to BVloc(Ω0) “conormally”, an assumption made p ..."
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Cited by 13 (4 self)
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A bstract. Let X be a bounded vector field with bounded divergence defined in an open set Ω of Rd, transverse to a hypersurface S. Let Ω0 be an open subset of Ω such that the Hausdorff measure Hd−1(Ω\Ω0) = 0. We suppose that the vector field X belongs to BVloc(Ω0) “conormally”, an assumption made precise in the text, which is satisfied whenever the gradients of the coefficients of X have locally only a single component which is actually a Radon measure. This class can be invariantly defined and contains the socalled piecewise W 1,1 functions studied in [Li]. We prove the uniqueness of L ∞ solutions for the Cauchy problem related to X across the hypersurface S. We use for the proof some simple arguments of geometric measure theory to get rid of closed sets with codimension> 1. Next, we need an anisotropic regularization argument analogous to the one used in [Bo]. C o ntent s 1. In t ro duct io n Our general framework. The one and two dimensional cases. The DiPernaLions ’ theorem on W 1,1 vector fields with bounded divergence. The ColombiniLerner’s result on BV vector fields with bounded divergence. 2. A n ew re su lt Definition 2.1. The class conormal BVloc. Theorem 2.4. Uniqueness of L ∞ solutions for a class of BV vector fields. 3. F ir st pa rt o f th e pro o f Step 1: nonnegative solutions are unique. Lemma 3.1.
Differential Equations with singular fields
"... Abstract. This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow ( ..."
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Cited by 12 (4 self)
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Abstract. This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to BV in dimension 2 and SBV in higher dimensions. 1
Uniqueness and nonuniqueness for nonsmooth divergence free transport
 Séminaire XEDP, Ecole Polytechnique
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THE FLOW ASSOCIATED TO WEAKLY DIFFERENTIABLE VECTOR FIELDS: RECENT RESULTS AND OPEN PROBLEMS
"... Abstract. We illustrate some recent developments of the theory of flows associated to weakly differentiable vector fields, listing the regularity/structural conditions considered so far, extensions to state spaces more general than Euclidean and open problems. Key words. Continuity equation, Transpo ..."
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Cited by 3 (1 self)
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Abstract. We illustrate some recent developments of the theory of flows associated to weakly differentiable vector fields, listing the regularity/structural conditions considered so far, extensions to state spaces more general than Euclidean and open problems. Key words. Continuity equation, Transport equation, Flow. AMS(MOS) subject classifications. 46E35, 34A12, 35R05.
unknown title
, 2003
"... www.elsevier.com/locate/cam Central schemes and systems of conservation laws with discontinuous coe%cients modeling gravity separation of polydisperse suspensions ..."
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www.elsevier.com/locate/cam Central schemes and systems of conservation laws with discontinuous coe%cients modeling gravity separation of polydisperse suspensions