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Geometric Integration Applied to Moving Mesh Methods and Degenerate Lagrangians


Tyranowski, Tomasz Michal (2014) Geometric Integration Applied to Moving Mesh Methods and Degenerate Lagrangians. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PH3X-YH23.


Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:geometric integration; variational integrators; multisymplectic integrators; moving mesh methods; moving mesh partial differential equations; differential-algebraic equations; Runge-Kutta methods; numerical analysis; geometric mechanics; field theory; solitons; Sine-Gordon equation;
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marsden, Jerrold E. (advisor)
  • Desbrun, Mathieu (advisor)
Thesis Committee:
  • Desbrun, Mathieu (chair)
  • Owhadi, Houman
  • Beck, James L.
  • Hou, Thomas Y.
Defense Date:26 September 2013
Record Number:CaltechTHESIS:12042013-185815472
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8038
Deposited By: Tomasz Tyranowski
Deposited On:16 Dec 2013 23:54
Last Modified:04 Oct 2019 00:03

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