Analysis of the heterogeneous multiscale method for ordinary differential equations
Analysis of the heterogeneous multiscale method for ordinary differential equations
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DOI：
10.4310/cms.2003.v1.n3.a3
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发表时间：
2003-09
影响因子：
1
通讯作者：
Weinan E
中科院分区：
文献类型：
--
作者：
Weinan E
研究了平均法[1]。这里假设f和g在φ中是周期性的，并且有界为ε → 0。在多尺度分析的标准术语中，我们称x和φ为这些系统的快变量，y和I为慢变量。但是，我们也有一个特别的兴趣，系统的快速和缓慢的变量存在，但不能事先明确确定。目前，还没有一个统一的战略来处理类型（1.1）和（1.2）的问题。然而，有大量的文献分别对刚性系统的类型（1.1）和振动系统的类型（1.2）。从Dahlquist和Gear [5]的开创性工作开始，人们在设计刚性常微分方程的有效数值方法[7]方面做了大量工作，例如向后微分公式，隐式Runge-Kutta方法[5]，外推方法和Rosenbrock方法[7，8，6]。关于振动系统也有大量的工作，其中一些是解析的[1]，一些是数值的[9]。我们将应用异构多尺度方法（HMM）的框架[3]，这是解决多尺度问题的通用方法。在两个尺度的情况下，宏观尺度和微观尺度，HMM由两个组成部分：选择一个传统的宏观尺度求解器，这里是一个标准的ODE求解器，并估计在宏观尺度求解器中使用的有效力，通过使用微尺度模型进行数值实验，并处理所获得的数据。如果系统具有两个以上的分离尺度，则可以迭代该过程。对于多时间尺度系统，特别是刚性常微分方程，有许多相关的数值方法。尽管刚性常微分方程最流行的数值方法来自隐式方法，如向后微分公式，但也提出了许多显式方法[8，2，7]。在最简单的版本中，这些方法本质上是Runge-Kutta，其中的每个阶段都是向前的。
studied in averaging methods [1]. Here f and g are assumed to be periodic in φ and bounded as ε → 0. In the standard terminology of multiscale analysis, we would call x and φ the fast variables of these systems, y and I the slow variables. But we also have a particular interest on systems for which the fast and slow variables exist but cannot be explicitly identified beforehand. At the present time, there does not exist a unified strategy for dealing with both problems of type (1.1) and (1.2). There is, however, a large literature on stiff systems of the type (1.1) and oscillatory systems of the type (1.2) separately. Starting from the pioneering work of Dahlquist and Gear [5], there has been extensive work on designing efficient numerical methods for stiff ODEs [7], such as the backward differentiation formula, implicit Runge-Kutta methods [5], extrapolation methods and Rosenbrock methods [7, 8, 6]. There is also extensive work on oscillatory systems, some of which are analytical [1], and some are numerical [9]. We will apply the framework of the heterogeneous multiscale method (HMM) [3], which is a general methodology for problems with multiscales. In the case of two scales, a macroscale and a microscale, HMM consists of two components: selection of a conventional macroscale solver, here a standard ODE solver, and estimating the effective forces used in the macroscale solver by performing numerical experiments using the microscale model and processing the data obtained. The procedure can be iterated if the system has more than two separated scales. There are a number of related numerical methods for systems with multiple time scales, in particular for stiff ODEs. Even though the most popular numerical methods for stiff ODEs stem from implicit methods such as the backward differentiation formula, a number of explicit methods have also been proposed [8, 2, 7]. In the simplest version, these methods are Runge-Kutta in nature, each stage of which is a forward