We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

我们考虑了一系列变分时间离散化，它们是不连续Galerkin（dG）和连续Galerkin–Petrov（cGP）方法的推广。该家族的特征在于两个参数。一个描述多项式ansatz阶，而另一个描述与全局平滑度相关联，该全局平滑度由子区间两端的高阶配置条件确保。应用于Dahlquist的稳定性问题，提出的方法提供了与dG或cGP方法相同的稳定性。假设使用合适的Hermite型正交规则来评估变分条件下的积分，则将变分时间离散化方法与特殊的搭配方法相联系。对于这种情况，我们提供误差估计，数值实验，