A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

Abstract

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.

References in corpus (6)

• Self-organization in systems of self-propelled particles
• A finite volume scheme for nonlinear degenerate parabolic equations
• Dimensionality of Local Minimizers of the Interaction Energy
• A new interaction potential for swarming models
• Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions
• Stationary States and Asymptotic Behaviour of Aggregation Models with Nonlinear Local Repulsion

Cited by in corpus (61)

• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
• Scaling Algorithms for Unbalanced Transport Problems
• Structure preserving schemes for nonlinear Fokker-Planck equations and applications
• Phase Transitions in a kinetic flocking model of Cucker-Smale type
• Fisher information regularization schemes for Wasserstein gradient flows
• Mean Field Limits for Interacting Diffusions in a Two-Scale Potential
• Biological Modelling with Nonlocal Advection Diffusion Equations
• The Filippov characteristic flow for the aggregation equation with mildly singular potentials
• Numerical Study of a Particle Method for Gradient Flows
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
• An accurate front capturing scheme for tumor growth models with a free boundary limit
• Convergence of a Fully Discrete and Energy-Dissipating Finite-Volume Scheme for Aggregation-Diffusion Equations
• Structure preserving schemes for the continuum Kuramoto model: phase transitions
• Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions
• Primal dual methods for Wasserstein gradient flows
• Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation
• First-order aggregation models with alignment
• Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
• Some Free Boundary Problems involving Nonlocal Diffusion and Aggregation
• Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation
• Positive and free energy satisfying schemes for diffusion with interaction potentials
• Well-balanced finite volume schemes for hydrodynamic equations with general free energy
• A Finite Volume Method for Continuum Limit Equations of Nonlocally Interacting Active Chiral Particles
• Swarm equilibria in domains with boundaries
• Large-Scale Wasserstein Gradient Flows
• Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
• Regularity of local minimizers of the interaction energy via obstacle problems
• Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit
• Entropy dissipation semi-discretization schemes for Fokker-Planck equations
• On a Class of Nonlocal Continuity Equations on Graphs
• Splitting Schemes & Segregation In Reaction-(Cross-)Diffusion Systems
• Entropic Wasserstein Gradient Flows
• Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations
• A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains
• Uniqueness of entire ground states for the fractional plasma problem
• Stationary States and Asymptotic Behaviour of Aggregation Models with Nonlinear Local Repulsion
• Multiple Patterns Formation for an Aggregation/Diffusion Predator-Prey System
• Structure-Preserving Numerical Methods for Nonlinear Fokker--Planck Equations with Nonlocal Interactions by an Energetic Variational Approach
• Field model for complex ionic fluids: analytical properties and numerical investigation
• A nonlocal-to-local approach to aggregation-diffusion equations
• Anisotropic long-range interaction investigated with cold atoms
• Phase transitions and bump solutions of the Keller-Segel model with volume exclusion
• Structure preserving schemes for Fokker-Planck equations with nonconstant diffusion matrices
• An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems
• Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system
• Multi-scale modeling of Snail-mediated response to hypoxia in tumor progression
• Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics
• A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration
• Structure preserving schemes for mean-field equations of collective behavior
• Discrete approximation of the minimizing movement scheme for evolution equations of Wasserstein gradient flow type with nonlinear mobility
• Modelling pattern formation through differential repulsion
• Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations
• Mean-field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods
• Drift-diffusion equations with saturation
• A phenotype-structured mathematical model for the influence of hypoxia on oncolytic virotherapy
• Degenerate Cahn-Hilliard systems: From nonlocal to local
• Relaxation limit of the aggregation equation with pointy potential
• A higher-order gradient flow scheme for a singular one-dimensional diffusion equation
• A unified structure preserving scheme for a multi-species model with a gradient flow structure and nonlocal interactions via singular kernels