Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method.

翻译：在求解初值问题时，跨方法并行时间积分可提供小规模并行性。采用对角扫描器的谱延迟校正（SDC）——该方法与Van der Houwen和Sommeijer提出的迭代Runge-Kutta方法密切相关——可利用的线程数等于底层配置法中的求积节点数。然而，其收敛速度、效率与稳定性关键取决于所用系数。先前方法采用数值优化来寻找合适参数。与之不同，我们提出一种解析求解最优参数的理论框架。研究表明，所得并行SDC方法具有与成熟串行SDC变体非常相似的稳定域和收敛阶。通过建立计算成本模型（假设并行SDC实现效率为80%），我们证明所提变体在性能上可与串行SDC、已发表的并行SDC系数、Picard迭代、显式RKM-4以及四阶对角隐式Runge-Kutta方法相竞争。