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

Area of Science:

• Computational mathematics
• Chemical kinetics
• Pattern formation

Background:

• Inhomogeneous steady states in reaction-diffusion equations are typically found using time-dependent simulations or nonlinear solvers.
• Standard nonlinear solvers often struggle with convergence, especially with sensitive initial guesses, leading to homogeneous solutions or divergence.
• Efficient and robust numerical methods are crucial for exploring spatial patterns across various parameter regimes.

Purpose of the Study:

• To develop a novel numerical approach for efficiently and robustly computing inhomogeneous steady states of nonlinear reaction-diffusion equations.
• To combine the robustness of temporal schemes with the fast convergence of nonlinear solvers.
• To provide a more reliable method for exploring spatial patterns in reaction-diffusion systems.

Main Methods:

• Introduction of the adaptive implicit Euler with inexact solver (AIIE) method.
• Application of AIIE to nonlinear reaction-diffusion equations in one, two, and three spatial dimensions.
• Comparative analysis of AIIE against traditional temporal schemes and nonlinear solvers (e.g., Newton's method).

Main Results:

• The AIIE method demonstrates significantly improved efficiency compared to standard temporal schemes.
• AIIE exhibits superior robustness in convergence compared to typical nonlinear solvers, reducing issues with initial guesses.
• The method successfully identified inhomogeneous spatial patterns across different dimensions and parameter spaces.

Conclusions:

• The AIIE method offers a more desirable approach for searching inhomogeneous spatial patterns in reaction-diffusion equations.
• This new method enhances the systematic numerical exploration of reaction-diffusion systems within large parameter spaces.
• AIIE provides a robust and efficient alternative for computational studies of pattern formation.