In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods and the technique of relaxation for numerical methods of ODEs. We explain the algorithm in detail and show numerical results for two- and three-derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained.

翻译：本文发展了一类能够离散地保持特定函数性质的高阶多导数时间积分方法。关键要素包括近期发展的Hermite-Birkhoff预估-校正方法以及常微分方程数值方法中的松弛技术。我们详细阐述了该算法，并展示了二导数和三导数方法的数值结果，将松弛与非松弛方法进行了比较。数值结果表明，通过付出松弛的轻微代价，即可获得改进的格式。