Abstract
dc:descriptionThis thesis covers numerical methods for discontinuous differential equations. In the context of ordinary differential equations, we introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order and conserved quantities across the interface. In the context of partial differential equations we introduce a numerical method based on discrete exterior calculus for the phase field equation. We prove the phase field variable remains bounded and satisfies mass conservation. Further, our method works on embedded two-dimensional Delaunay meshes in \mathbb{R}3. Numerical examples on an embedded cylinder in \mathbb{R}3 demonstrate both boundedness and mass conservation up to machine precision.
Degree
thesis:*- Name thesis:degree_name
- Ph.D.
- Level thesis:degree_level
- Dissertation
- Discipline thesis:degree_discipline
- Mathematics
- Grantor
- University of Illinois at Urbana-Champaign
- Year dc:date
- 2022
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Wojtalewicz, Nikolas
- Contributors dc:contributor
-
- Hirani, Anil
- Wan, Andy
- DeVille, Lee
- Laugesen, Richard
Subjects
dc:subject × 4Rights
dc:rights- Statement dc:rights
-
- Copyright 2022 Nikolas Wojtalewicz
- Language dc:language
- en, eng
Identifiers
dc:identifier.*- Handle dc:identifier
- https://hdl.handle.net/2142/116225