A novel, high-order, parallel-in-time integration procedure for complex fluid flows

在这个项目中，我们分析了一类常微分方程组数值处理的隐式多重导数格式，并将其推广到复偏微分方程组的时间并行积分器中。MD格式具有较高的精度和相对较低的存储要求，为与不连续Galerkin谱元素(DGSEM)的空间离散相结合提供了一个非常有吸引力的选择。以预测器/校正器的形式编写的MD格式的子集提供了自然的时间并行化的可能性，当然对高性能计算是有吸引力的。本项目的第一步是分析和开发新的MD格式，特别关注内存需求、离散化误差和时间自适应。有了这些计划，它们就与地方政府经济发展部结合在一起。在这里，焦点集中在多导数项的一致离散化，这对于稳定的方法以及对DGSEM的节点结构的开发是必不可少的。最后，对MD方案的时间并行性能进行了研究。由于实现是在适用于高性能计算的空间并行代码框架内完成的，因此不仅可以进行分析研究，而且可以解决真实世界示例中的实际问题。

In this project, the class of implicit multiderivative (MD) schemes for numerical treatment of ordinary differential equations is analyzed, extended and used as a parallel-in-time integrator for complex partial differential equations. Yielding high-order of accuracy with relatively low storage requirements, MD schemes offer a highly attractive option for being combined with a spatial discretization by discontinuous Galerkin spectral elements (DGSEM). A subset of the class of MD schemes, written in predictor/corrector form, provides a natural possibility of time parallelization, being of course attractive for high performance computing.The first step in this project is the analysis and development of novel MD schemes with a special focus on memory requirements, discretization error and temporal adaptivity. Having those schemes in place, they are combined with the DGSEM. Here, focus is on a consistent discretization of the multiderivative terms being essential for a stable method as well as on the exploitation of the nodal structure of the DGSEM. Finally, the time-parallel capabilities of MD schemes are investigated. As the implementation is done within a spatially parallelized code framework suitable for high performance computing, not only analytical investigations are carried out but also practical questions on real-world examples can be tackled.