Taking into account the limited accuracy of the energy-preserving exponential integrator of order two (Li and Wu, 2016 [29]) for conservative or dissipative systems with highly oscillatory solutions, this paper is devoted to presenting a uniform framework to design energy-preserving exponential integrators of arbitrarily high order based on the modifying integrator theory. To this end, we first show that the second-order energy-preserving exponential integrator is a B-series method. Using the adapted substitution law, we then prove that there exist arbitrary order energy-preserving exponential integrators and show how to design arbitrarily high-order integrators by finding the truncated modified differential equations. As an example, the fourth-order energy-preserving exponential integrator is constructed in detail. The stability and convergence of the proposed integrators are analyzed as well. Finally, numerical experiments are accompanied, including both ODEs and PDEs, and the numerical results demonstrate the remarkable superiority over the existing energy-preserving integrators for highly oscillatory systems in the literature.

考虑到具有高振动解的保守或耗散系统的二阶能量指数积分器的精度有限（Li和Wu，2016 [29]），本文致力于提供一个统一的框架来设计能量节约基于修正积分理论的任意高阶指数积分器。为此，我们首先证明二阶节能指数积分器是B系列方法。然后，使用自适应替换定律，证明存在任意阶的能量守恒指数积分器，并说明如何通过查找截断的修正微分方程来设计任意高阶积分器。例如，详细构造四阶节能指数积分器。还分析了所提出的积分器的稳定性和收敛性。最后，还进行了包括ODE和PDE在内的数值实验，数值结果表明，相对于文献中现有的用于高振荡系统的节能积分器，其优越性明显。