解决子网格微尺度相互作用以离散化随机偏微分方程
Resolve subgrid microscale interactions to discretise stochastic partial differential equations
AI总结：
本文利用随机中心流形理论，通过将物理域划分为重叠有限元，解析子网格微尺度噪声与非线性动力学的耦合，为随机偏微分方程提供粗粒度空间离散化方法。
摘要：
构建随机偏微分方程的离散模型非常微妙。随机中心流形理论为非线性随机偏微分或差分方程（如随机强迫Burgers方程）的粗粒度、宏观空间离散化提供了新颖支持。将物理域划分为有限长度的重叠单元，使该方法能够解析相邻单元之间完全耦合的动力学相互作用。该方法的关键在于，基础理论组织了通过相邻单元内及单元间非线性动力学相互作用的巨量子网格微尺度噪声过程的解析。具有跨有限元粗结构的噪声过程对离散模型最为重要。它们的影响也会扩散开来，使空间离散化中的噪声弱相关。非线性相互作用还有两个进一步的影响：加性强迫在离散化中产生乘性噪声；并且在宏观离散化中有效出现新的噪声过程。这里开发的技术和理论可广泛应用于许多耗散随机偏微分和差分方程的合理离散化。
英文摘要：
Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial differential or difference equations such as the example of the stochastically forced Burgers' equation. Dividing the physical domain into finite length overlapping elements empowers the approach to resolve fully coupled dynamical interactions between neighbouring elements. The crucial aspect of this approach is that the underlying theory organises the resolution of the vast multitude of subgrid microscale noise processes interacting via the nonlinear dynamics within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in the spatial discretisation. Nonlinear interactions have two further consequences: additive forcing generates multiplicative noise in the discretisation; and effectively new noise processes appear in the macroscale discretisation. The techniques and theory developed here may be applied to soundly discretise many dissipative stochastic partial differential and difference equations.
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