耗散常微分方程的高阶数值均匀化方法
High Order Numerical Homogenization for Dissipative Ordinary Differential Equations
AI总结：
本文提出一种针对含双时间尺度耗散常微分方程的高阶数值均匀化方法，基于异构多尺度方法框架，通过迭代公式构造高阶微观求解器，实现任意阶精度提升，并数值验证了有效性。
摘要：
我们提出了一种针对包含两个时间尺度的耗散常微分方程（ODE）的高阶数值均匀化方法。本质上，只能全局地推导出一阶均匀化模型。为了实现高阶方法，我们必须在异构多尺度方法（HMM）框架内采用数值方法。通过逐步细化的微观求解器，在输入数据足够光滑的条件下，可以实现任意阶的精度提升。基于我们推导的高阶微观求解器的公式，我们提出了一种迭代公式来计算微观求解器。利用该迭代公式，我们开发了一种高效实用的实现方法。最后，我们给出了几个数值算例来验证新模型和数值方法的有效性。
英文摘要：
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high order method, we have to adopt a numerical approach in the framework of the heterogeneous multiscale method (HMM). By a successively refined microscopic solver, the accuracy improvement up to arbitrary order is attained providing input data smooth enough. Based on the formulation of the high order microscopic solver we derived, an iterative formula to calculate the microscopic solver is then proposed. Using the iterative formula, we develop an implementation to the method in an efficient way for practical applications. Several numerical examples are presented to validate the new models and numerical methods.
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