In this article, we present the Python framework pySDC for solving collocation problems with spectral deferred correction (SDC) methods and their time-parallel variant PFASST, the parallel full approximation scheme in space and time. pySDC features many implementations of SDC and PFASST, from simple implicit timestepping to high-order implicit-explicit or multi-implicit splitting and multilevel SDCs. The software package comes with many different, preimplemented examples and has seven tutorials to help new users with their first steps. Time parallelism is implemented either in an emulated way for debugging and prototyping or using MPI for benchmarking. The code is fully documented and tested using continuous integration, including most results of previous publications. Here, we describe the structure of the code by taking two different perspectives: those of the user and those of the developer. The first sheds light on the front-end, the examples, and the tutorials, and the second is used to describe the underlying implementation and the data structures. We show three different examples to highlight various aspects of the implementation, the capabilities, and the usage of pySDC. In addition, couplings to the FEniCS framework and PETSc, the latter including spatial parallelism with MPI, are described.

We present a novel space-time parallel version of the Barnes-Hut tree code PEPC using PFASST, the Parallel Full Approximation Scheme in Space and Time. The naive use of increasingly more processors for a fixed-size N-body problem is prone to saturate as soon as the number of unknowns per core becomes too small. To overcome this intrinsic strong-scaling limit, we introduce temporal parallelism on top of PEPC's existing hybrid MPI/PThreads spatial decomposition. Here, we use PFASST which is based on a combination of the iterations of the parallel-in-time algorithm parareal with the sweeps of spectral deferred correction (SDC) schemes. By combining these sweeps with multiple space-time discretization levels, PFASST relaxes the theoretical bound on parallel efficiency in parareal. We present results from runs on up to 262,144 cores on the IBM Blue Gene/P installation JUGENE, demonstrating that the spacetime parallel code provides speedup beyond the saturation of the purely space-parallel approach.