aspg-34-4479-1791482242 20261008175722 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Prospects for Applied Mathematics and Data Analysis PAMDA 2836-4449 2026 6 1 Sergey Drominko Faculty of Information Technology and Robotics, Vitebsk State Technological University, Belarus Erina Kovachiskaya Faculty of Information Technology and Robotics, Vitebsk State Technological University, Belarus 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. 2026 15 22 4479 https://creativecommons.org/licenses/by/4.0/ 10.54216/PAMDA.060103 https://www.americaspg.com/journal/34/article/4479 https://www.americaspg.com/storage/articles/manuscripts/01KZSAFBTBCEYFJG37GYH48S39.pdf