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

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.