We present a novel class of integrators for differential equations that are suitable for parallel in time computation, whose structure can be considered as a generalization of the extrapolation methods. Starting with a low order integrator (preferably a symmetric second order one) we can build a set of second order schemes by few compositions of this basic scheme that can be computed in parallel. Then, a proper linear combination of the results (obtained from the order conditions associated to the corresponding Lie algebra) allows us to obtain new higher order methods. In this letter we present the structure of the methods, how to obtain several methods, we notice some order barriers that depend on the structure of the compositions used and finally, we show how this analysis can be further carried to obtain new and higher order schemes.

我们提出了一类适用于并行时间计算的微分方程积分器，其结构可以被认为是外推方法的推广。从低阶积分器（最好是对称二阶积分器）开始，我们可以通过可以并行计算的基本方案的几个组合来构建一组二阶方案。然后，结果的适当线性组合（从与相应李代数关联的阶条件获得）允许我们获得新的高阶方法。在这封信中，我们介绍了方法的结构，如何获得几种方法，我们注意到一些依赖于所用组合物结构的阶障碍，最后，我们展示了如何进一步进行这种分析以获得新的和更高阶的方案.