Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport

Daniel Matthes, Eva-Maria Rott, Giuseppe Savaré, André Schlichting

TL;DR

This work develops a structure-preserving spatial discretization for the fourth-order Derrida–Lebowitz–Speer–Spohn (DLSS) equation on the circle, grounded in a novel diffusive transport gradient-flow framework. It introduces a discrete diffusive transport distance and a mobility-based finite-volume discretization that preserves positivity, mass, entropy dissipation, Fisher information dissipation, and Hellinger contractivity, while delivering a rigorous convergence result to a weak DLSS solution as the mesh is refined. The approach unifies continuous and discrete gradient-flow perspectives (Wasserstein and diffusive transport) and provides strong a priori estimates and a robust convergence analysis. The resulting scheme offers a rigorous, structure-preserving tool for simulating DLSS dynamics, with potential implications for quantum drift-diffusion models and interface-fluctuation phenomena.

Abstract

We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies an a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.

A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport

TL;DR

This work develops a structure-preserving spatial discretization for the fourth-order Derrida–Lebowitz–Speer–Spohn (DLSS) equation on the circle, grounded in a novel diffusive transport gradient-flow framework. It introduces a discrete diffusive transport distance and a mobility-based finite-volume discretization that preserves positivity, mass, entropy dissipation, Fisher information dissipation, and Hellinger contractivity, while delivering a rigorous convergence result to a weak DLSS solution as the mesh is refined. The approach unifies continuous and discrete gradient-flow perspectives (Wasserstein and diffusive transport) and provides strong a priori estimates and a robust convergence analysis. The resulting scheme offers a rigorous, structure-preserving tool for simulating DLSS dynamics, with potential implications for quantum drift-diffusion models and interface-fluctuation phenomena.

Abstract

We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies an a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.
Paper Structure (22 sections, 17 theorems, 212 equations, 2 figures)

This paper contains 22 sections, 17 theorems, 212 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. DLSS equation --- origins and fundamental analytical results
  3. DLSS equation --- structural properties
  4. DLSS equation --- numerical schemes
  5. Diffusive transport on the continuous torus
  6. Gradient flows with respect to the diffusive transport distance
  7. Diffusive transport on the discrete torus
  8. Discretization and notation
  9. Discrete optimal diffusive connection
  10. Discrete diffusive transport distance
  11. Gradient flows with respect to the discrete diffusive transport distance
  12. Variational discretization of the DLSS equation
  13. Mobility function
  14. Properties of the discretization
  15. Proof of Lemma \ref{['lem:dDLSSpos']}
  16. ...and 7 more sections

Key Result

Lemma 1

For any given $\mu^0,\mu^1\in\mathcal{P}({{\mathbb S}^1})$, there exists a connecting curve $(\mu^s,{\mathfrak m}^s)_{s\in[0,1]}\in\mathcal{CE}(\mu^0,\mu^1)$ with In particular, any two measures in $\mathcal{P}({{\mathbb S}^1})$ can be connected by a curve of action less than one.

Figures (2)

  • Figure 1: Logarithmic plot of numerical solution to \ref{['eq:dDLSS0']} started from a discretization of $\bar{\rho}(x) = Z_{m,\varepsilon}^{-1} (*){\varepsilon^{1/2}+ (*){\frac{1+\cos(2\pi x)}{2}}^{\!m}}^{\!2}$ with normalization constant $Z_{m,\varepsilon}$ such that $\bar{\rho}\in \mathcal{P}({{\mathbb S}^1})$, and parameters $\varepsilon=0.001$, $m=1,2,8,16$ (top left, top right, bottom left, bottom right).
  • Figure 2: Semi-logarithmic plot of discretized Lyapunov functions $\mathcal{H}$, $\mathcal{F}$, $\mathbb{H}(\cdot,1)$, and $\mathcal{L}$ for the initial datum from Figure \ref{['fig:diffusive']} with $m=16$ (left: initial time interval; right: convergence till machine precision with asymptotic exponential rate $-12.2$).

Theorems & Definitions (36)

  • Lemma 1: Comparison with $\dot H^{-2}({{\mathbb S}^1})$-norm
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Definition 4: Weak convergence in ${\mathcal{C}}$
  • Definition 5: Admissible mobility
  • Lemma 6: Lower semi-continuity of the action
  • proof
  • ...and 26 more