[Paper Review] Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations

Luís Almeida, Federica Bubba+2 more|arXiv (Cornell University)|Mar 28, 2018

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TL;DR

This paper presents two novel implicit finite-volume schemes for the Fokker-Planck and Keller-Segel equations that preserve energy dissipation, mass conservation, positivity, and steady-state structure at the discrete level. By leveraging gradient flow and Scharfetter-Gummel discretization strategies with upwinding, the schemes ensure well-posedness and stability without time-step restrictions, enabling accurate simulation of pattern formation and distinction between physical instabilities and numerical artifacts.

ABSTRACT

The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes whose goals are to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach. These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting feature is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotony properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare.

Motivation & Objective

• To develop numerical schemes that preserve fundamental physical properties—energy dissipation, mass conservation, positivity, and steady states—for the Fokker-Planck and Keller-Segel equations.
• To address numerical challenges in simulating pattern formation due to strong nonlinearities and advection terms, especially distinguishing physical Turing instabilities from numerical artifacts.
• To construct implicit time discretizations that avoid CFL restrictions while maintaining all key discrete properties.
• To compare the performance of gradient flow-based and Scharfetter-Gummel-based schemes in capturing complex patterns accurately.
• To prove well-posedness and monotonicity-based convergence of the schemes using subsolution-supersolution arguments.

Proposed method

• The schemes are derived from two distinct symmetrization strategies: the gradient flow structure and the Scharfetter-Gummel method for drift-diffusion equations.
• Upwinding is incorporated into both schemes to ensure preservation of positivity, mass conservation, and energy dissipation at the semi-discrete level.
• The fully discrete schemes are constructed using implicit time stepping, ensuring no time-step restriction and unconditional stability.
• Monotonicity properties, analogous to the maximum principle, are used to prove existence and uniqueness of solutions via subsolution-supersolution theory.
• The discrete energy dissipation is rigorously proven by constructing a Lyapunov functional that mirrors the continuous energy inequality.
• The schemes are validated numerically, with comparisons showing superior accuracy in resolving steady states and transient patterns over standard upwind methods.

Experimental results

Research questions

• RQ1 Can implicit finite-volume schemes preserve energy dissipation, mass conservation, positivity, and steady-state structure in the Fokker-Planck and Keller-Segel equations?
• RQ2 How do gradient flow-based and Scharfetter-Gummel-based discretizations compare in preserving the qualitative behavior of solutions with strong nonlinearities?
• RQ3 To what extent can upwinding alone ensure the preservation of key physical properties at the semi-discrete level?
• RQ4 Can implicit time discretization eliminate the CFL condition while maintaining all discrete conservation and stability properties?
• RQ5 How can numerical artifacts such as spurious oscillations or false steady states be distinguished from physical pattern formation in these systems?

Key findings

• The proposed schemes preserve energy dissipation at both semi-discrete and fully discrete levels, with the energy functional decreasing monotonically over time.
• Both schemes maintain strict positivity of the solution and conserve total mass, even for non-smooth or peaked initial data.
• The schemes are unconditionally stable and well-posed, with existence and uniqueness of solutions proven via subsolution-supersolution theory.
• The Scharfetter-Gummel-based scheme provides a robust alternative to the gradient flow approach, particularly effective for general sensitivity functions φ(u).
• Numerical simulations demonstrate that the schemes accurately capture Turing-type instabilities and peak-like steady states, avoiding spurious oscillations common in standard upwind schemes.
• The implicit time discretization allows for large time steps without loss of stability or accuracy, making the method suitable for long-time simulations of pattern formation.