# Finite Volume Method for Nonlinear Nonlocal Equations¶

## 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:

$\partial_t\rho_i = \nabla\cdot\Big[\rho_i\nabla\Big(H'(\rho)+V(\mathbf{x})+W_{ii}\ast\rho_i+\sum\limits_{j\neq{i}}{W_{ij}\ast\rho_j}+\epsilon\rho\Big)\Big], \mathbf{x}\in\mathbb{R}^2, t\gt{0}$

$$\mathbf{\rho_i}$$ : 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 . 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 .

System Requirements

Software Requires MATLAB Release Compatibility
MATLAB Created with R2018a

## Support¶

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