使用微观模拟器的大规模解中间隙-齿方案的高阶精度
Higher order accuracy in the gap-tooth scheme for large-scale solutions using microscopic simulators
AI总结：
提出一种多尺度计算框架，通过将微观模拟器限制在域的小块（齿）中，并利用插值估计宏观场（间隙），实现宏观尺度的高阶精度建模，重点解决齿的边界条件以实现高阶一致性。
摘要：
我们正在开发一个多尺度计算框架，该框架使得“微观”描述层次的模型（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。计划是将微观规则限制在域的小块（即“齿”）中，然后通过插值来估计“间隙”中的宏观场。这里面临的挑战是找到微观模拟器块的通用边界条件，这些条件能够适当地连接广泛分离的“齿”，从而在宏观尺度上实现高阶精度。我们首先在微观模拟器是偏微分方程的典型示例的最简单情况下探索这些问题。对于这种情况，解析解提供了比较。我们认为经典的高阶插值提供了块边界条件，这些条件在间隙-齿方案中实现了任意高阶的一致性，并且经过谨慎处理后数值稳定。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：首先使用整体离散化的动力系统方法；其次通过选定数值问题的特征值。当应用于微观模拟块时，这些块边界条件应能实现高效的宏观尺度模拟。
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 the ``macroscopic'' length scales of interest. The plan is to use the microscopic rules restricted to small patches of the domain, the ``teeth'', followed by interpolation to estimate macroscopic fields in the ``gaps''. The challenge addressed here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated ``teeth'' to achieve high order accuracy over the macroscale. Here we start exploring the issues in the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. For this case analytic solutions provide comparisons. We argue that classic high-order interpolation provides patch boundary conditions which achieve arbitrarily high-order consistency in the gap-tooth scheme, and with care are numerically stable. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly using the dynamical systems approach of holistic discretisation; and secondly through the eigenvalues of selected numerical problems. When applied to patches of microscopic simulations these patch boundary conditions should achieve efficient macroscale simulation.
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