通过微观模拟确定“粗方程”的性质：婴儿水方案
Deciding the Nature of the "Coarse Equation" through Microscopic Simulations: the Baby-Bathwater Scheme
摘要：
多尺度计算的最新发展允许微观/随机演变粒子分布的预期宏观行为的“粗方程”的解，而无需以封闭形式获得这些粗方程。通过适当初始化的微观模拟脉冲，“按需”获得闭合。在这种方法中，微观模拟器与宏观行为的有效耦合需要对不可用的“粗糙方程”的性质做出某些决定。这样的判定包括（a）确定方程中有效的最高空间导数，（b）粗略方程是否满足某些守恒定律，以及（c）粗略动力学是否是哈密尔顿的。这些决定影响边界条件的数量和类型，以及在良好的解决方案实践中采用的算法的性质。在没有一个明确的公式的时间导数，我们提出，实施和验证一个简单的计划，用于决定这些和其他类似的问题，只使用微观模拟器的粗方程。微观模拟周期性边界条件下进行适当选择的家庭的随机初始条件;评估样本方差的某些统计数据的模拟合奏，使我们能够推断出最高阶的空间导数活跃在粗糙的方程。本着同样的精神，我们展示了如何确定是否存在某种粗糙的守恒定律，我们讨论了存在一个粗糙的哈密顿量或可积性的可积性测试。我们认为，这样的计划构成了多尺度计算的方程自由的方法的重要组成部分。
英文摘要：
Recent developments in multiscale computation allow the solution of ``coarse equations'' for the expected macroscopic behavior of microscopically/stochastically evolving particle distributions without ever obtaining these coarse equations in closed form. The closure is obtained ``on demand'' through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macrosocopic behavior embodied in this approach requires certain decisions about the nature of the unavailable ``coarse equation''. Such decisions include (a) the determination of the highest spatial derivative active in the equation, (b) whether the coarse equation satisfies certain conservation laws, and (c) whether the coarse dynamics is Hamiltonian. These decisions affect the number and type of boundary conditions as well as the nature of the algorithms employed in good solution practice. In the absence of an explicit formula for the temporal derivative, we propose, implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Microscopic simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We argue that such schemes constitute an important part of the equation-free approach to multiscale computation.
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