用于无方程建模的通用齿边界条件
General tooth boundary conditions for equation free modelling
AI总结：
提出一种通过微观模拟补丁（齿）间的高阶插值边界条件实现宏观尺度高效建模的框架，并在线性偏微分方程上验证了任意高阶一致性。
摘要：
我们正在开发一个多尺度计算框架，该框架使得在“微观”描述层次上的模型（例如格子玻尔兹曼、蒙特卡洛或分子动力学模拟器）能够在感兴趣的“宏观”长度尺度上执行建模任务。该计划是使用仅限于域的小“补丁”（即“齿”）的微观规则，并通过插值来桥接“间隙”。在这里，我们探索了耦合广泛分离的微观模拟“齿”的通用边界条件，这些条件在宏观尺度上实现了高阶精度。我们展示了最简单的情况，即微观模拟器是偏微分方程的典型例子。我们认为，宏观场的经典高阶插值为微观模拟器所需的任何边界条件提供了正确的强制项。这种插值导致了齿边界条件，该条件实现了任意高阶一致性。通过两种方式在一类线性偏微分方程上证明了高阶一致性：首先通过选定数值问题的方案特征值；其次使用整体离散化的动力系统方法处理一类通用的线性PDE。解析建模表明，对于一大类微观系统，子网格场和有效宏观模型在很大程度上独立于齿大小和特定的齿边界条件。当应用于微观模拟的补丁时，这些齿边界条件有望实现高效的宏观尺度模拟。我们预期相同的方法也将准确耦合更高空间维度中的补丁模拟。
英文摘要：
We are developing a framework for multiscale computation which enables models at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators, to perform modelling tasks at ``macroscopic'' length scales of interest. The plan is to use the microscopic rules restricted to small "patches" of the domain, the "teeth'', using interpolation to bridge the "gaps". Here we explore general boundary conditions coupling the widely separated ``teeth'' of the microscopic simulation that achieve high order accuracy over the macroscale. We present the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. We argue that classic high-order interpolation of the macroscopic field provides the correct forcing in whatever boundary condition is required by the microsimulator. Such interpolation leads to Tooth Boundary Conditions which achieve arbitrarily high-order consistency. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly through the eigenvalues of the scheme for selected numerical problems; and secondly using the dynamical systems approach of holistic discretisation on a general class of linear \textsc{pde}s. Analytic modelling shows that, for a wide class of microscopic systems, the subgrid fields and the effective macroscopic model are largely independent of the tooth size and the particular tooth boundary conditions. When applied to patches of microscopic simulations these tooth boundary conditions promise efficient macroscale simulation. We expect the same approach will also accurately couple patch simulations in higher spatial dimensions.
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