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Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

Anastasiia Hraivoronska, André Schlichting, Oliver Tse

TL;DR

This work develops a variational framework for the semi-discrete Scharfetter--Gummel scheme applied to the aggregation-diffusion equation, establishing a tilt-independent generalized gradient structure that persists for both $\epsilon>0$ and $\epsilon=0$. It proves discrete-to-continuum convergence (EDP convergence) by deriving a $\Gamma$-limit for the Fisher information and showing compactness of density-flux pairs, ultimately recovering the Otto gradient flow for the continuum problem. The vanishing-diffusion analysis links the SG scheme to the upwind scheme and, in the limit, to the aggregation equation, both at the discrete and continuous levels, via energy-dissipation balance and gradient-flow convergence. The results provide a robust, structure-preserving route from finite-volume approximations to continuum gradient-flow solutions, with implications for numerical analysis of drift-diffusion-type systems and collective-behavior models. Overall, the paper demonstrates that variational convergence and tilt-independence are key to overcoming discretization-induced artifacts and ensuring convergence to well-posed gradient-flow limits across diffusion regimes.

Abstract

In this paper, we explore the convergence of the semi-discrete Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume space discretization that works consistently for any non-negative diffusion constant. This allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously. The zero-diffusion limit for the Scharfetter-Gummel scheme corresponds to the upwind finite volume scheme for the aggregation equation. In both cases, we establish a convergence result in terms of gradient structures, recovering the Otto gradient flow structure for the aggregation-diffusion equation based on the 2-Wasserstein distance.

Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

TL;DR

This work develops a variational framework for the semi-discrete Scharfetter--Gummel scheme applied to the aggregation-diffusion equation, establishing a tilt-independent generalized gradient structure that persists for both and . It proves discrete-to-continuum convergence (EDP convergence) by deriving a -limit for the Fisher information and showing compactness of density-flux pairs, ultimately recovering the Otto gradient flow for the continuum problem. The vanishing-diffusion analysis links the SG scheme to the upwind scheme and, in the limit, to the aggregation equation, both at the discrete and continuous levels, via energy-dissipation balance and gradient-flow convergence. The results provide a robust, structure-preserving route from finite-volume approximations to continuum gradient-flow solutions, with implications for numerical analysis of drift-diffusion-type systems and collective-behavior models. Overall, the paper demonstrates that variational convergence and tilt-independence are key to overcoming discretization-induced artifacts and ensuring convergence to well-posed gradient-flow limits across diffusion regimes.

Abstract

In this paper, we explore the convergence of the semi-discrete Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume space discretization that works consistently for any non-negative diffusion constant. This allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously. The zero-diffusion limit for the Scharfetter-Gummel scheme corresponds to the upwind finite volume scheme for the aggregation equation. In both cases, we establish a convergence result in terms of gradient structures, recovering the Otto gradient flow structure for the aggregation-diffusion equation based on the 2-Wasserstein distance.
Paper Structure (22 sections, 30 theorems, 309 equations, 3 figures)

This paper contains 22 sections, 30 theorems, 309 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. Assumptions on tessellations
  4. Assumptions on potentials
  5. Main results
  6. Gradient Structures: Discrete and Continuous
  7. Otto-Wasserstein gradient structure for diffusion-type equations
  8. Generalized gradient structure for finite volume schemes
  9. A new gradient structure for the Scharfetter--Gummel scheme
  10. Variational Convergence for the Tilt-Independent Structure
  11. Compactness
  12. Γ-convergence of the Fisher information
  13. EDP convergence
  14. Vanishing Diffusion Limit
  15. Discrete Case
  16. ...and 7 more sections

Key Result

Theorem A

Let $\{(\mathcal{T}_h,\Sigma_h)\}_{h>0}$ be a family of tessellations satisfying ass:tessellation, and assume ass:pointy-potential to hold for the interaction potential $W$. Further, let $\{(\rho^h,j^h)\}_{h>0}$ be a family of GGF-solutions eq:SG with initial data $\{\rho_{\text{in}}^h\}_{h>0}$ havi Then there exists a (not relabelled) subsequence of admissible continuous reconstructions $\{(\hat{

Figures (3)

  • Figure 1: We are interested in the convergence results indicated by the arrows in the sense of EDP convergence. The arrows with the label "$h\to 0$" indicate the convergence of the discrete approximation to the continuous problem. The arrows with the label "$\epsilon\to 0$" indicate the vanishing diffusion limit.
  • Figure 2: This diagram depicts the main results of this paper. The generalized gradient structure for the Scharfetter--Gummel scheme is established in Lemma \ref{['lem:flux-force-for-tilt']}. The labels on the arrows indicate the corresponding convergence statements in the sense of the EDP convergence.
  • Figure 3: Consider the 2-d Cartesian grid. Let the central cell be the cell $K$ and $w$ be a vector starting at $x_K$. Then the line orthogonal to $w$ (dashed line) splits the set of vectors $\{(x_L - x_K)\}_{L\in\mathcal{T}^h_K}$ into two groups: one such that $w\cdot (x_L - x_K) > 0$ (i.e. $\{he_1, -he_2\}$) and the other such that $w\cdot (x_L - x_K) < 0$, (i.e, $\{-he_1, he_2\}$).

Theorems & Definitions (69)

  • Remark 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3: GF solutions
  • ...and 59 more