Enhancing the Nektar++ spectral/hp element framework for parallel-in-time simulations

We describe the efficient implementation of the Parareal algorithm in the Nektar++ software, an open-source spectral/hp element framework for the solution of partial differential equations, which has been designed to achieve high-scalability on high-performance computing (HPC) clusters using distributed parallelism. Recently, time-parallel integration techniques are being recognized as a potential solution to further increase concurrency and computational speed-up beyond the limits of strong scaling obtained from a pure spatial domain decomposition. Amongst the various time-parallel approaches proposed in the literature, the Parareal algorithm is a non-intrusive and iterative approach, exploiting a fine and a coarse solvers to achieve time-parallelism, and can be applied to both linear and non-linear problems. We discuss the details of the implementation and discuss the specific techniques used to adapt the code to a time-parallel framework. We demonstrate the application of these methods to multiple linear and non-linear problems provided by the existing Nektar++ solvers.