整体离散化确保二维空间动力学保真度
Holistic discretisation ensures fidelity to dynamics in two spatial dimensions
AI总结：
利用中心流形理论，提出一种整体离散化方法，通过系统解析亚网格微观动力学，为反应扩散问题构建渐近精确的宏观闭包，并应用于Ginzburg-Landau方程。
摘要：
动力系统理论的发展为偏微分方程及其他微观系统的离散化提供了新支持。通过系统解析亚网格微观动力学，新方法为偏微分方程的离散模型构建了渐近精确的宏观闭包。本文探讨二维空间中的反应扩散问题。中心流形理论确保了慢流形、整体离散化的存在性、快速吸引性以及系统近似性。有限元的特殊耦合确保了所得离散化与偏微分方程在任意高阶上保持一致。计算机代数处理了巨大的代数细节，如具体应用于Ginzburg-Landau方程所示。然而，二维中的高阶模型似乎需要一种混合数值与代数方法，本文也对此进行了开发。由于由方程残差驱动，本文的建模方法可轻松适用于多空间维度中广泛类别的反应扩散微分方程和格点方程。
英文摘要：
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of discrete models of the \pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the \pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.
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