Finite Volume Method for Nonlinear Nonlocal Equations


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 [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


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