#### Publication Date

Spring 5-15-2020

#### Abstract

As the clock speeds of individual processors level off and the amount of parallel resources continue to increase rapidly, further exploitation of parallelism is necessary to improve compute times. For time-dependent differential equations, the serial computation of time-stepping presents a bottleneck, but parallel-in-time integration methods offer a way to compute the solution in parallel along the time domain. Parallel-in-time methods have been successful in achieving speedup when computing solutions for parabolic problems; however, for problems with large hyperbolic terms and no strong diffusivity, parallel-in-time methods have traditionally struggled to offer speedup. While work has been done to understand why parallel-in-time methods struggle to converge quickly for hyperbolic problems, a few parallel-in-time techniques have been demonstrated to achieve speedup for certain hyperbolic problems. We consider a previously proposed technique based on parareal, which is a general parallel-in-time method that uses a relatively cheap coarse-grid approximation to compute error corrections to accelerate the solution of a fine-grid time-marching problem. In particular, we look at a method which constructs an asymptotically time-averaged approximation on the parareal coarse grid, which has been shown to work well when solving hyperbolic problems whose solutions exhibit fast oscillations in the time dimension. Using the generalizability of the parareal method into the multigrid-reduction-in-time (MGRIT) algorithm, we investigate the expansion of the two-grid asymptotic parareal method to a multilevel MGRIT setting. In particular, we research runtime improvements when rapid oscillations are present by using the multilevel capabilities and FCF-relaxation smoothing aspects of MGRIT. Methods to improve compute speed in flow regimes without fast temporal oscillations are also examined.

#### Degree Name

Mathematics

#### Level of Degree

Masters

#### Department Name

Mathematics & Statistics

#### First Committee Member (Chair)

Jacob Bayer Schroder

#### Second Committee Member

Jehanzeb Hameed Chaudhry

#### Third Committee Member

Rob Falgout

#### Fourth Committee Member

Deborah Sulsky

#### Language

English

#### Keywords

multigrid, parallel-in-time, multiscale, shallow water equations, parareal, multigrid reduction in time

#### Document Type

Thesis

#### Recommended Citation

Abel, Nicholas. "Multilevel Asymptotic Parallel-in-Time Techniques For Temporally Oscillatory PDEs." (2020). https://digitalrepository.unm.edu/math_etds/173