Published

A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems.

Wing-Cheong Lo, Long Chen, Ming Wang, and Qing Nie

Journal of Computational Physic. 231(15), 5062-5077, 2012.

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ABSTRACT:

 
An inhomogeneous steady state pattern of nonlinear
reaction-diffusion equations with no-flux boundary conditions is
usually computed by solving the corresponding time-dependent
reaction-diffusion equations using temporal schemes.  Nonlinear
solvers (e.g., Newton's method) take less CPU time in direct
computation for the steady state; however, their convergence is
sensitive to the initial guess, often leading to divergence or
convergence to spatially homogeneous solution. Systematically
numerical exploration of spatial patterns of reaction-diffusion
equations under different parameter regimes requires that the
numerical method be efficient and robust to initial condition or
initial guess, with better likelihood of convergence to an
inhomogeneous pattern.  Here, a new approach that combines the
advantages of temporal schemes in robustness and Newton's method in
fast convergence in solving steady states of reaction-diffusion
equations is proposed. In particular, an adaptive implicit Euler with
inexact solver (AIIE) method is found to be much more efficient than
temporal schemes and more robust in convergence than typical nonlinear
solvers (e.g., Newton's method) in finding the inhomogeneous pattern.
Application of this new approach to two reaction-diffusion equations
in one, two, and three spatial dimensions, along with direct
comparisons to several other existing methods, demonstrates that AIIE
is a more desirable method for searching inhomogeneous spatial
patterns of reaction-diffusion equations in a large parameter space.