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A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
José A. Carrillo, Alina Chertock, Yanghong Huang
TL;DR
Nonlinear nonlocal equations have multiple steady states and challenging behaviors, including singular interactions and different equilibration scales. The paper develops a positivity-preserving finite-volume method with discrete free-energy dissipation and tests it across stationary states, long-time dynamics, singular kernels, and two-dimensional problems. The experiments demonstrate that the scheme handles non-smooth states, metastable time scales, concentrations, and self-similar behavior, while also being used to explore regimes beyond current theoretical knowledge.
Problem
Nonlinear nonlocal equations can have multiple steady states, singular interactions, non-smooth states, and long-time behaviors that require numerical study beyond currently available theoretical knowledge.
Method
The paper proposes a positivity-preserving finite-volume scheme for the general nonlinear nonlocal PDE and constructs a discrete free energy dissipated by its semi-discrete formulation.
Results
The numerical experiments show accurate treatment of stationary states and long-time asymptotics, including non-smooth states, multiple time scales, concentrations, and self-similar behavior from singular kernels.
Takeaways & Limitations
The scheme is used to investigate stationary states, equilibration, Keller–Segel regimes, aggregation equations, singular kernels, and two-dimensional numerical asymptotics.
Abstract
from arXiv · showhide
We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.
1 Introduction
The paper studies nonlinear nonlocal equations with gradient-flow structure and develops a positivity-preserving, entropy-decreasing finite-volume method. Numerical experiments assess stationary states, equilibration, singular kernels, and multidimensional behavior.
- Motivation: The equations model interacting systems including granular flows, porous-medium flows, Fokker–Planck equations, chemotaxis, and collective biological behavior.The interaction potential may range from singular Newtonian kernels to smooth power-law kernels.
- Gradient-flow structure: The associated free energy combines internal, potential, and interaction energies, while its time derivative defines an entropy dissipation functional.This gradient-flow structure motivates entropy-based numerical design.
- Motivation: Steady states can be multiple and are often studied under assumptions on their support and characterizing equation.Their explicit forms are available only for particular interaction potentials.
- Method: The proposed finite-volume method preserves non-negativity and dissipates a discrete free energy for the semi-discrete scheme.The method covers nonlocal terms in one and two dimensions and generalizes easily to unstructured meshes at first order.
- Numerical experiments: The experiments study spatial convergence, stationary-state computation, equilibration rates, Keller–Segel regimes, aggregation with singular kernels, and two-dimensional asymptotics.The study includes non-smooth and discontinuous steady states as well as different time scales.
2 Numerical Method
The paper develops one- and two-dimensional finite-volume schemes for nonlinear nonlocal equations, proving positivity preservation and discrete entropy dissipation under suitable time discretization and CFL conditions.
- Scope: The proposed methods cover one- and two-dimensional uniform-mesh finite-volume discretizations, with extensions to higher dimensions and non-uniform structured meshes described as straightforward.The analysis includes error estimates and convergence results.
- Reconstruction and fluxes: Upwind numerical fluxes approximate the continuous flux, while discrete velocities are computed from the confinement and interaction potentials using second-order quadrature.The quadrature error is O(∆x^2) for the regular contributions, with singular kernels requiring special treatment.
- Reconstruction and fluxes: Piecewise linear reconstructions are combined with slope limiting so reconstructed point values remain nonnegative when cell averages are nonnegative.A generalized minmod limiter is used in the numerical experiments, with θ = 2 controlling numerical viscosity.
- Singular kernels: For locally integrable singular kernels, the first interaction integral can retain O(∆x^2) accuracy, whereas a kernel W(x) ∼ |x|^-α may yield O(∆x^(2−α)) error in the nearby-cell correction.Symmetry makes the corresponding self-cell correction vanish even when W is singular.
- Implementation: Numerical examples use a third-order SSP-RK solver, while the discrete convolution is identified as the computational bottleneck and can be accelerated with fast convolution algorithms.The second-order method reduces to first order when piecewise constant reconstruction is used.
- Structure preservation: The one-dimensional scheme preserves nonnegative cell averages under a CFL condition, using forward Euler or higher-order SSP time integration.The proof represents updated cell averages as linear combinations of nonnegative reconstructed point values.
- Structure preservation: The scheme dissipates a discrete entropy, satisfying a discrete analogue of the continuous inequality that the entropy derivative is bounded above by negative discrete dissipation.The result is established with no-flux boundary conditions.
- Two-dimensional extension: The two-dimensional scheme preserves nonnegative cell averages under a modified CFL condition and exhibits an analogous dissipative property.The positivity result applies when forward Euler or an SSP Runge-Kutta method is used for time integration.
3 Numerical Experiments
The numerical experiments test convergence, equilibration, stationary-state selection, critical-mass behavior, singular kernels, and two-dimensional dynamics. They show that the scheme captures non-smooth states, multiple time scales, blow-up or diffusion regimes, and mesh-sensitive effects.
- Steady states: Spatial Order and Time Stabilization: The practical steady-state error is O(∆x^α) in L∞ and O(∆x^(α+1)) in L1 when the exact state is C^α-Hölder continuous.Even on a coarse grid with ∆x = 2/5, the numerical steady state agrees well with the exact one except near the boundary.
- Steady states: Spatial Order and Time Stabilization: Attraction can produce arbitrarily slow equilibration at large distances, followed by exponential decay once separated density bumps approach each other.For m = 3, σ = 1, and ν = 1.48, the experiments display these two distinct time scales.
- Steady states: Spatial Order and Time Stabilization: Different initial supports can converge to different stationary states, while entropy decreases substantially mainly when the solution topology changes.For small mass, asymmetric stationary states may have nonzero basins of attraction, although the symmetric state is the global free-energy minimizer.
- Generalized Keller-Segel model: The generalized Keller-Segel experiments distinguish diffusion-dominated, balanced, and aggregation-dominated regimes through compact states, critical-mass transitions, and coexistence of diffusion and blow-up.For m > (d − α)/d, compact stationary states are expected; equality identifies a critical mass, while lower values permit both behaviors.
- Generalized Keller-Segel model: For subcritical mass, self-similar solutions converge exponentially toward equilibrium, while the profiles concentrate toward a Dirac Delta as M approaches Mc.The reported decay rate is independent of mass and is exactly O(e−2t) in the cited two-dimensional Keller-Segel comparison.
- Aggregation equations and two-dimensional simulations: Singular repulsive-attractive kernels and two-dimensional interactions expose numerical artifacts and complex dynamics, including clump formation, overshoots, and concentration on lower-dimensional sets.Mid-point quadrature is oscillation free, whereas exact-integral evaluation can overshoot near support boundaries; numerical diffusion is used to suppress this effect.