Flux-Corrected Diagonal Frog: second order and positivity at all time steps

2026-07-22

AI Summary

This work addresses the challenge that linear second-order finite difference schemes for the Fokker–Planck equation struggle to simultaneously preserve solution positivity and achieve second-order accuracy, often requiring restrictive time-step constraints. The authors propose a nonlinear flux-corrected finite difference (FCDF) method that decomposes a second-order directional operator into a monotone M-matrix core and an antidiffusive correction term. Within an implicit time-stepping framework, a Zalesak-type flux limiter is iteratively applied, activating only in unresolved layers to ensure unconditional positivity, exact mass conservation, and global second-order L₁ convergence for arbitrary time steps. By combining an active-set semismooth Newton solver with defect-correction time integration, the approach successfully restores full second-order temporal accuracy, as demonstrated on Ornstein–Uhlenbeck and convection-dominated test problems.

Abstract

By Godunov's theorem, linear second-order finite-difference schemes for the Fokker-Planck equation cannot preserve positivity. The Diagonal Frog (DF) framework previously bypassed this barrier using eventual positivity, but required a strict minimum time step. This paper resolves the small-step limitation using a nonlinear extension of the DF solvers. We split the second-order directional operator into a monotone M-matrix core and an antidiffusive flux correction. A Zalesak-type limiter is then applied iteratively within the implicit banded solve. The resulting Flux-Corrected DF (FCDF) schemes (variants A and B) are unconditionally positive across all time steps. Because the limiter acts on fluxes rather than point values, these schemes conserve discrete mass exactly and maintain second-order accuracy. Crucially, the limiter activates only within unresolved layers. This ensures the global $L_1$ convergence remains second-order uniformly in the cell Péclet number, avoiding the first-order degradation seen in the Chang-Cooper scheme. The method's Picard iteration is contractive under a purely convective step restriction. To support arbitrary step sizes, we develop an active-set reformulation. This solves the system using a semismooth Newton iteration, where computational cost scales only with the number of nodes where positivity binds. Finally, we introduce a defect-corrected time stepping approach that restores second-order time accuracy. Numerical experiments on Ornstein-Uhlenbeck and advection-dominated benchmarks confirm our claims.

Problem

• Fokker-Planck equation
• positivity preservation
• second-order accuracy
• flux correction
• time step restriction

Innovation

• Flux-Corrected Diagonal Frog
• positivity-preserving
• second-order accuracy
• Fokker-Planck equation
• Zalesak limiter