[Paper Review] Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations

Jingwei Hu, Xiangxiong Zhang|arXiv (Cornell University)|Mar 31, 2021

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TL;DR

This paper presents two semi-implicit finite difference schemes—second-order and fourth-order accurate in space—for the Fokker-Planck and Keller-Segel equations. The schemes are positivity-preserving and energy-dissipative, with the fourth-order scheme achieving these properties under mild time step and mesh constraints, marking the first high-order spatial discretization to simultaneously preserve positivity and energy decay.

ABSTRACT

In this work, we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker-Planck equation and the Keller-Segel equations. The two proposed schemes are first order accurate in time, explicitly solvable, and second order and fourth order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proven positivity-preserving and energy-dissipative: the second order scheme can achieve so unconditionally; the fourth order scheme only requires a mild time step and mesh size constraint. Furthermore, the fourth order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.

Motivation & Objective

• To develop structure-preserving numerical schemes for gradient flow equations like Fokker-Planck and Keller-Segel with long-time stability.
• To ensure discrete solutions remain non-negative and energy decays monotonically, mimicking the continuous variational structure.
• To construct high-order spatial discretizations (second and fourth order) that maintain positivity and energy dissipation without restrictive time steps.
• To enable accurate long-time simulations and reliable steady-state computation for chemotaxis and diffusion models.

Proposed method

• A semi-implicit first-order time discretization is used, solving the system in a fully implicit or semi-implicit manner to ensure stability.
• Second- and fourth-order spatial accuracy are achieved via finite difference approximations of the continuous finite element method with linear and quadratic polynomials on uniform meshes.
• The schemes are derived from the equivalent variable-coefficient diffusion form of the continuity equation: ∂ₜρ = ∇·(ℳ∇(ρ/ℳ)), where ℳ = exp(−(𝒱 + 𝒲∗ρ)).
• Positivity is enforced through a monotonicity condition on the discrete scheme, proven via discrete maximum principle arguments.
• Energy dissipation is established by showing the discrete free energy functional decreases over time, mirroring the continuous energy decay.
• The fourth-order scheme requires only a mild time step and mesh size constraint (lower bound on Δt) for monotonicity, not an upper bound.

Experimental results

Research questions

• RQ1 Can a high-order finite difference scheme preserve both positivity and energy dissipation for the Fokker-Planck and Keller-Segel equations?
• RQ2 What time and spatial discretization constraints are necessary to ensure monotonicity and positivity in high-order schemes?
• RQ3 How do the proposed schemes perform in long-time simulations and near blow-up regimes?
• RQ4 Can the fourth-order scheme maintain positivity and energy decay even when mesh constraints are relaxed?

Key findings

• The second-order scheme is unconditionally positivity-preserving and energy-dissipative, ensuring stability without time step restrictions.
• The fourth-order scheme achieves positivity and energy dissipation under a mild constraint: the time step must be bounded below, not above.
• Numerical results show both schemes converge to steady states with ∥ρⁿ⁺¹ − ρⁿ∥∞ ≤ 10⁻⁸ by T ≈ 13.52 for subcritical initial mass.
• For supercritical initial mass, the fourth-order scheme captures the blow-up dynamics more accurately than the second-order scheme, especially at T = 0.8.
• Energy evolution plots confirm monotonic decay in both schemes, even after blow-up, with no violation of energy dissipation or positivity.
• The fourth-order scheme is the first high-order spatial discretization to simultaneously preserve positivity and energy dissipation, enabling accurate long-time simulations.