Date of Award

Summer 2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Nail Yamaleev

Committee Member

Ruhai Zhou

Committee Member

Fang Hu

Committee Member

Andrey Chernikov

Abstract

An accurate prediction of unsteady physical phenomena arising in various applications (e.g., rotorcraft and turbomachinery flows, fluid-structure interaction, maneuvering flight conditions, etc.) requires a very large number of time steps, thus considerably increasing the total computational time, because conventional time integrators are inherently sequential. Parallel-in-time methods offer a promising direction for drastically reducing the computational time and achieving such scalability that is required for solving these unsteady problems on modern supercomputers with hundreds of thousands of computing cores. The parallel performance of existing parallel-in-time algorithms for nonlinear equations especially of the hyperbolic or mixed type is far from being satisfactory. To address this problem, we propose a new method for parallelization of implicit schemes for nonlinear partial differential equations of arbitrary type. The time derivative term is discretized by using the method of lines based on the implicit first-order backward difference (BDF1) scheme, while the inviscid and viscous terms are approximated by using central finite difference discretizations. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. This all-at-once system at each Newton iteration is block bidiagonal, which can be inverted directly in a blockwise manner, thus leading to a set of fully decoupled equations associated with each time level. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations. In contrast to the existing parallel-in-time algorithms, the proposed method preserves the quadratic rate of convergence of the Newton method of the corresponding sequential scheme and provides a nearly ideal speedup on up to 32 computing cores for nonlinear PDEs with both smooth and discontinuous solutions. To improve the scalability, we combine the proposed method with the Parareal algorithm, which allows us to further increase the speedup by a factor of 2-3 for the 2-D nonlinear heat and Burgers equations. These novel parallel-in-time methods can be directly combined with spatial domain-decomposition algorithms, thus demonstrating a strong potential for obtaining much higher speedups on large computer platforms as compared with the current state-of-the-art methods based on parallelization of the spatial discretization alone.

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DOI

10.25777/3xpz-r440

ISBN

9798384456629

ORCID

0009-0005-5254-4311

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Available for download on Tuesday, October 07, 2025

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