Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme
Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme
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通过微观模拟确定粗略方程的性质：婴儿洗澡水方案
DOI：
10.1137/070692303
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
I. Kevrekidis
中科院分区：
文献类型：
--
作者：
Ju Li;P. Kevrekidis;C. Gear;I. Kevrekidis
多尺度计算的最新发展允许微观演变粒子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。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。这样的决定包括（a）在粗略方程中有效的最高空间导数，（B）方程是否满足某些守恒定律，或者（c）粗略动力学是哈密顿的还是耗散的。这些决定影响边界条件的数量和类型以及所采用的算法。在没有一个明确的公式的时间导数，我们提出，实施，并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。适当选择的随机初始条件的家庭进行定期边界条件下的模拟;评估样本方差的某些统计数据的模拟合奏允许我们推断最高阶的空间导数活跃在粗方程。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无方程方法的重要组成部分。
Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles 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 macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation. Such decisions include (a) the highest spatial derivative active in the coarse equation, (b) whether the equation satisfies certain conservation laws, or (c) whether the coarse dynamics is Hamiltonian or dissipative. These decisions affect the number and type of boundary conditions as well as the algorithms employed. 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. 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 believe that such schemes constitute an important part of the equation-free approach to multiscale computation.