R-Adaptive Multisymplectic and Variational Integrators
- Creators
- Tyranowski, Tomasz M.
- Desbrun, Mathieu
Abstract
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 paper, 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. Numerical results for the Sine–Gordon equation are also presented.
Additional Information
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Received: 30 May 2019; Revised: 3 July 2019; Accepted: 9 July 2019; Published: 18 July 2019. Author Contributions: Our contributions were equally balanced in a true collaboration. This research received no external funding. We would like to extend our gratitude to Michael Holst, Eva Kanso, Patrick Mullen, Tudor Ratiu, Ari Stern, and AbigailWacher for the useful comments and suggestions. We are particularly indebted to Joris Vankerschaver and Melvin Leok for their support, discussions, and interest in this work. We dedicate this paper in memory of Jerrold E. Marsden, who began this project with us. The authors declare no conflict of interest.Attached Files
Published - mathematics-07-00642.pdf
Submitted - 1303.6796.pdf
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Additional details
- Eprint ID
- 97244
- Resolver ID
- CaltechAUTHORS:20190718-142449959
- Created
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2019-07-18Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field