Volume 6 • Issue 1 • PP: 15–22 • 2026
A Symmetric Discrete-Gradient Finite-Volume Method for Positive Drift–Diffusion Flows
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© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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Consider the periodic gradient flow ∂tu = ∂x(m(u)∂xμ) , μ = h′(u)+V, E [u] = Z 1 0 {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 M 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.
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