Parallel Time Integration Methods for Hyperbolic PDEs in Application

Abstract

Parallel in Time (PinT) integration methods are iterative methods which seek to increase computational parallelism in numerical solutions to initial value problems by solving future time statesconcurrently. PinT methods have been successfully applied to problems with sufficient dissipation, but struggle to converge quickly for problems without much dissipation or that are dominated by convection or hyperbolic in nature. In these cases, PinT is not expected to work well. Because of this, PinT methods have not been not favored by many computational scientists, though we may still desire faster time to solution in convective problems. In this dissertation, we push the usefulness of PinT methods to include hyperbolic PDEs. To do so, we present a modification for PinT methods which computes a time averaged state rather than a pointwise in time quantity. Time averaged quantities like drag and lift of an airfoil are useful to engineers in application, so we ask how well PinT can be applied in computing them. The calculation of time averages has not seen any research in the PinT community. We hope that this research expands the uses for PinT methods by exhibiting its usefulness in computing useful quantities for PDEs beyond those of parabolic behavior.