Multimethods for the Efficient Solution of Multiscale Differential Equations

Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive. The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze. The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Computer Science and Applications

Department dc:contributor.department

Computer Science

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2021

Author dc:creator

• Roberts, Steven Byram

Chair dc:contributor.committeechair

• Sandu, Adrian

Committee members dc:contributor.committeemember

• Warburton, Timothy
• Woodward, Carol S.
• Cao, Young
• Ribbens, Calvin J.

• Time integration
• Multirate
• Implicit-Explicit
• Runge–Kutta
• General Linear Method

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:32417

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/104872