Discontinuous differential equations

Wojtalewicz, Nikolas

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https://hdl.handle.net/2142/116225 Copy

Description

Title

Discontinuous differential equations

Author(s)

Wojtalewicz, Nikolas

Issue Date

2022-07-14

Director of Research (if dissertation) or Advisor (if thesis)

• Hirani, Anil
• Wan, Andy

Doctoral Committee Chair(s)

DeVille, Lee

Committee Member(s)

Laugesen, Richard

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Numerical Analysis
• Ordinary Differential Equations
• Partial Differential Equations
• Numerical Methods

Abstract

This 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.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Nikolas Wojtalewicz

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