The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation

Gianazza, Ugo; Savaré, Giuseppe; Toscani, Giuseppe

doi:10.1007/s00205-008-0186-5

Cited by 130 publications

(183 citation statements)

References 34 publications

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“…Let (T h ) h>0 be a sequence of admissible meshes satisfying (15) and (ρ h , V h , u h ) h>0 be a sequence of approximate solutions given by the scheme (16)- (19). Then there exists ( ρ, V , u) such that, up to a subsequence, ρ h → ρ, V h → V and u h → u strongly in L 2 (Ω) as h → 0, and ( ρ, V , u) is a weak solution to the system (4)-(8) satisfying ρ ≥ ρ > 0 in Ω.…”

Section: Theoremmentioning

confidence: 99%

“…( K∈T by solving scheme (19). We need to prove that the mapping T is well defined by establishing that each of the above steps admits a unique solution and that T maps U into itself.…”

Section: Existence Of a Discrete Solutionmentioning

confidence: 99%

“…We consider the case u D σ = 0 for all σ ∈ E D K,ext , multiply (19) by u K , and sum over K ∈ T . Then…”

Section: Existence Of a Discrete Solutionmentioning

confidence: 99%

“…Existence results for the one-dimensional transient equations can be found in [24,25]. The multi-dimensional case (without pressure and electric field) was analyzed in [19,23].…”

Section: Introductionmentioning

confidence: 99%

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A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Chainais-Hillairet

Gisclon

Jüngel

2010

Numer. Methods Partial Differential Eq.

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Abstract. A finite-volume scheme for the stationary unipolar quantum drift-diffusion equations for semiconductors in several space dimensions is analyzed. The model consists of a fourth-order elliptic equation for the electron density, coupled to the Poisson equation for the electrostatic potential, with mixed Dirichlet-Neumann boundary conditions. The numerical scheme is based on a Scharfetter-Gummel type reformulation of the equations. The existence of a sequence of solutions to the discrete problem and its numerical convergence to a solution to the continuous model are shown. Moreover, some numerical examples in two space dimensions are presented.

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Section: Theoremmentioning

confidence: 99%

“…( K∈T by solving scheme (19). We need to prove that the mapping T is well defined by establishing that each of the above steps admits a unique solution and that T maps U into itself.…”

Section: Existence Of a Discrete Solutionmentioning

confidence: 99%

“…We consider the case u D σ = 0 for all σ ∈ E D K,ext , multiply (19) by u K , and sum over K ∈ T . Then…”

Section: Existence Of a Discrete Solutionmentioning

confidence: 99%

“…Existence results for the one-dimensional transient equations can be found in [24,25]. The multi-dimensional case (without pressure and electric field) was analyzed in [19,23].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Chainais-Hillairet

Gisclon

Jüngel

2010

Numer. Methods Partial Differential Eq.

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show abstract

“…[3,6,8,9,15,19,26,37], just to name a few), and therefore any method that uses only the properties of this structure has the potential of application to a wide range of problems. Consequently, our approach here is to limit our use of information to those properties that follow directly from the gradient-flow structure.…”

Section: Introductionmentioning

confidence: 99%

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

et al. 2011

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We study a singular-limit problem arising in the modelling of chemical reactions. At finite ε > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier et al. (SIAM J Math Anal, 42(4):1805-1825, 2010, using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the ε-problem converge to a solution of the limiting problem.

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Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

Jüngel¹,

Milišić

2014

Numerical Methods Partial

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New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non-negativity and the entropy dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G-stability theory with entropy dissipation methods and to introduce a nonlinear transformation of variables, which provides a quadratic structure in the equations. It is shown that G-stability of the one-leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross-diffusion system from population dynamics and a nonlinear fourth-order quantum diffusion model, for which the existence of semidiscrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second-order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results.

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The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation

Cited by 130 publications

References 34 publications

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

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