A Symmetric Discrete-Gradient Finite-Volume Method for

Positive Drift–Diffusion Flows

Sergey Drominko1,* Erina Kovachiskaya1

1 Faculty of Information Technology and Robotics, Vitebsk State Technological University, Belarus

Received: September 24, 2025 Revised: November 10, 2025 Accepted: January 06, 2026 ⋆ Corresponding author

ABSTRACT

Consider the periodic gradient flow

∂tu = ∂x(m(u)∂xμ) , μ = h′(u)+V, E [u] =

Z 1

{h(u)+Vu} dx,

for strictly positive density u, convex entropy density h, and mobility m > 0. We couple a centered finite-volume flux

with a symmetric two-state approximation of the chemical potential. Its entropy component is the divided difference

Dh(a,b) =

h(a)−h(b)

a−b

, Dh(a,a) = h′(a),

which enforces the discrete chain rule exactly. With midpoint edge mobility and the logarithmic state un+1

i =expzn+1

i ,

the nonlinear update is self-adjoint and, for every solved algebraic step, satisfies

E n+1

h −E n

h = −ΔtΣi

n+1/2

i+1/2

(μn+1/2

i+1 −μn+1/2

i )2

Δx

≤ 0.

The flux telescopes to conserve mass, the logarithmic parametrization confines finite roots to the positive cone, and

states satisfying h′(ui)+Vi = const are fixed points. A midpoint expansion gives second-order consistency in time;

the centered flux gives the same order in space. For h(u) = u(logu−1), a manufactured heat-flow calculation gives

observed L2 orders 1.998 and 1.999 under coupled refinement. In a confining Fokker–Planck test, the maximum

mass defect is 3.20×10−14 and the energy identity is satisfied within 1.42×10−14. Replacing Dh by the midpoint

chemical potential in the same one-step problem produces an energy-balance defect 1.60×10−2, isolating the role

of the temporal discrete gradient.

Keywords: Discrete gradient Finite volume method Drift–diffusion equation Gradient flow Free-energy dissipation

Positivity Time symmetry

1. INTRODUCTION

Nonlinear drift–diffusion equations often carry more mathematical

structure than their parabolic form alone suggests.

For a density u > 0 on the one-dimensional torus T = R/Z,