一种求解带小参数的常微分方程的简化多尺度法
A Simplified Multiscale Method for Solving Ordinary Differential Equations with Small Parameters

针对带小参数的微分方程求数值解已经有了很多种数值解法，比如多尺度法(Multiscale Methods)，将微分方程按照时间尺度进行划分，通过分开求解不同时间尺度下的子方程从而求解原方程。但是这种方法划分的尺度过于臃肿，极大的增加了运算时间；其次需要手动处理尺度方程，来避免久期项(secular terms)的影响导致最终求得的数值解有一定的误差。本文提出了一种改进版的多尺度法(Reductive Multiscale Methods)，将时间尺度的划分极大地简化，其次利用欧拉公式和泰勒展开的性质将久期项(secular terms)化到尺度方程内部，从而避免了久期项(secular terms)对数值解的影响。最后将该方法举例得到的数值解与多尺度法(Multiscale Methods)对比，在一定程度下，验证了改进算法的运算量小、高效率的优势。
There are many numerical solutions available for solving differential equations with small parame-ters, such as the Multiscale Methods, which divide the differential equations into time scales and solve the original equation by solving sub equations at different time scales separately. However, the scale of this method is too cumbersome, greatly increasing the computational time. Secondly, it is necessary to manually process the scale equation to avoid the influence of the duration term, which may lead to certain errors in the final solution. This article proposes an improved version of the Reduced Multiscale Methods, which greatly simplifies the division of time scales. Secondly, Tay-lor expansion and Euler formula are used to transform the duration terms into the scale equation, thereby avoiding the influence of the duration terms on the numerical solution. Finally, the numer-ical solution obtained by this method was compared with the multiscale methods, thereby verifying to some extent the advantages of the improved algorithm in terms of low computational complexity and high efficiency.