Fourier spectral/hp element method: investigation of time-stepping and parallelisation strategies

Abstract

As computer hardware has evolved, the time required to perform numerical simulations has reduced, allowing investigations of a wide range of new problems. This thesis focuses on algorithm optimization, to minimize run-time, when solving the incompressible Navier-Stokes equations. Aspects affecting performance related to the discretisation and algorithm parallelization are investigated in the context of high-order methods. The roles played by numerical approximations and computational strategies are highlighted and it is recognized that a versatile implementation provides additional benefits, allowing an ad-hoc selection of techniques to fit the needs of heterogeneous computing environments.

We initially describe the building blocks of a spectral/hp element and pure spectral method and how they can be encapsulated and combined to create a 3D discretisation, the Fourier spectral/hp element method. Time-stepping strategies are also described and encapsulated in a flexible framework based on the General Linear Method. After implementing and validating an incompressible Navier-Stokes solver, two canonical turbulent flows are analyzed.

Afterward a 2D hyperbolic equation is considered to investigate the efficiency of low- and high-order methods when discretising the spatial and temporal derivatives. We perform parametric studies, monitoring accuracy and CPU-time for different numerical approximations. We identify optimal discretisations, demonstrating that high-order methods are the computationally fastest approach to attain a desired accuracy for this problem.

Following the same philosophy, we investigate the benefits of using a hybrid parallel implementation. The message passing model is introduced to parallelize different kernels of an incompressible Navier-Stokes solver. Monitoring the parallel performance of these strategies the most efficient approach is highlighted. We also demonstrate that hybrid parallel solutions can be used to significantly extend the strong scalability limit and support greater parallelism.