A deterministic approximation of diffusion equations using particles
SIAM Journal on Scientific and Statistical Computing
An analysis of 1-D smoothed particle hydrodynamics kernels
Journal of Computational Physics
Particle methods for dispersive equations
Journal of Computational Physics
A general deterministic treatment of derivatives in particle methods
Journal of Computational Physics
Simulation of anisotropic diffusion by means of a diffusion velocity method
Journal of Computational Physics
ADI schemes for higher-order nonlinear diffusion equations
Applied Numerical Mathematics
Finite element approximation of a nonlinear cross-diffusion population model
Numerische Mathematik
On a practical implementation of particle methods
Applied Numerical Mathematics - Selected papers from the first Chilean workshop on numerical analysis of partial differential equations (WONAPDE 2004)
Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms
Applied Numerical Mathematics
Mathematics and Computers in Simulation
On a cross-diffusion population model deduced from mutation and splitting of a single species
Computers & Mathematics with Applications
Hi-index | 0.00 |
The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rachford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.