这篇文章主要跟随Marlis Hochbruck(2010) 的综述性文章《Exponential integrators》的逻辑，较为系统地讨论了指数时间差分方法的构造、分析、实现 和应用，其中主要讨论了该方法在具有光滑解的抛物型问题和一阶高振荡问 题中的实现和收敛性。
在抛物型问题中，对于线性问题可以直接使用指数积分法，即指数形式 的Euler 积分法、中点法和梯形法等方法；对于半线性问题则考虑使用类似于 Runge-Kutta 法的构造，在本文中构造并分析了指数Euler 法，然后提出了高 阶指数Runge-Kutta 法的收敛条件和一些样例。
在高振荡问题中，则主要考虑较为简单的一阶高振荡问题，并以Magnus积分法为引导，具体地分析了指数中点法的收敛条件和收敛性。
最后，在简单的一维刚性问题上实践了二阶和四阶的指数Runge-Kutta方法，并在不同步长下分析了两种方法的收敛性与收敛速度，发现指数时间差分方法确实具有较好的稳定性和收敛速度，并且考虑具体方法的构造仍然有非常大的发展前景，因此该方法是值得学习和进一步研究的。

Following the logic of the review article <Exponential integrators> by Marlis Hochbruck (2010), this paper systematically discusses the construction, analysis, implementation and application of the exponential time differencing method, and mainly discusses the implementation and convergence of the method in parabolic problems with smooth solutions and first-order high oscillation problems.
In parabolic problems, the exponential quadrature method can be directly used for linear problems, such as the exponential Euler integration method, the midpoint method and the trapezoidal method. For Semilinear problems, the construction similar to the Runge Kutta method is considered. In this paper, the exponential Euler method is constructed and analyzed, and then the convergence conditions and some examples of the higher-order exponential Runge Kutta method are proposed.
In high oscillation problems, the simple first-order high oscillation problem is mainly considered. Guided by the Magnus integral method, the convergence condition and convergence of the exponential midpoint method are analyzed in detail.
Finally, the second-order and fourth-order exponential Runge Kutta methods are applied to a simple one-dimensional stiff problem, and the convergence and convergence rate of the two methods are analyzed under different time length. It is foundthat the exponential time difference method does have good stability and convergence rate, and considering the construction of specific methods, it still has a great development prospect, Therefore, this method is worth learning and further research.