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首页> 外文期刊>Journal of Computational Physics >A robust and efficient method for steady state patterns in reaction-diffusion systems
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A robust and efficient method for steady state patterns in reaction-diffusion systems

机译：反应扩散系统稳态模式的鲁棒高效方法

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摘要

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.
机译：通常，通过使用时间方案求解相应的时变反应扩散方程，可以计算出无通量边界条件的非线性反应扩散方程的非均匀稳态模式。非线性求解器（例如牛顿法）在直接计算稳态时所需的CPU时间更少;但是，它们的收敛对最初的猜测很敏感，常常导致发散或收敛到空间均匀解。在不同参数范围内对反应扩散方程的空间模式进行系统的数值探索，要求数值方法对于初始条件或初始猜测是有效且稳健的，并且更有可能收敛到非均匀模式。在这里，提出了一种新的方法，该方法结合了鲁棒性的时间方案和快速收敛的牛顿法在求解反应扩散方程稳态下的优点。特别地，发现非均匀求解器的自适应隐式欧拉（AIIE）方法比时间方案效率更高，并且在收敛方面比典型的非线性求解器（例如，牛顿法）更鲁棒，可以找到不均匀的模式。将该新方法应用于一个，两个和三个空间维度中的两个反应扩散方程，并与其他几种现有方法进行直接比较，证明AIIE是一种更理想的方法，用于搜索反应扩散方程中的非均匀空间模式较大的参数空间。

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