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 |
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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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