# Finite Volume Method for Nonlinear Nonlocal Equations¶

## Documentations¶

## Overview¶

**Model Description:** For N groups in a 2D space, let \(\rho_i(\mathbf{x}, t)\) denotes the mass density distribution of group *i*
at time *t*, \(i=1,2,...,N\). The problem we are interested in:

\(\mathbf{\rho_i}\) : mass density of group *i*

\(\mathbf{\rho}\): \(\sum{\rho_i}\) , sum of densities of all N groups

\(\mathbf{H(\rho)}\): density of internal energy

\(\mathbf{V(\mathbf{x})}\): environmental confinement potential

\(\mathbf{W_{ii}}\): self-interaction potential (intraspecific interaction potential)

\(\mathbf{W_{ij}}(j\neq{i})\): cross-interaction potential (interspecific interaction potential)

\(\mathbf{\epsilon}\): diffusion coefficient

with initial condition \(\rho_i(\mathbf{x},0)=\rho_{i0}(\mathbf{x})\).

**Numeric Analysis:** This numeric scheme is developed based on the finite volume method described in Links [2]. The paper proposes a
finite method scheme for nonlinear nonlocal system for a single group in one and two spacial dimensions. We extend the scheme to include more than one group,
with cross-interaction mechanism mentioned in Links [3].

System Requirements

Software Requires | MATLAB Release Compatibility |
---|---|

MATLAB | Created with R2018a |

## Links¶

- Source Code: Github
- A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure: arXiv:1402.4252v2
- Zoology of a Nonlocal Cross-Diffusion Model for Two Species: Permalink

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