R-Adaptive Multisymplectic and Variational Integrators

Tomasz M Tyranowski,Mathieu Desbrun

doi:10.3390/math7070642

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.

Highlights

The purpose of this work is to design, analyze, and implement variational and multisymplectic integrators for Lagrangian partial differential equations with space-adaptive meshes
Moving mesh methods are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations
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

Summary

Introduction

The purpose of this work is to design, analyze, and implement variational and multisymplectic integrators for Lagrangian partial differential equations with space-adaptive meshes. We combine geometric numerical integration and r-adaptive methods for the numerical solution of Partial Differential Equations (PDEs). We show that these two fields are compatible, mostly due to the fact that, in r-adaptation, the number of mesh points remains constant and we can treat them as additional pseudo-particles of which the dynamics are coupled to the dynamics of the physical field of interest. Geometric (or structure-preserving) integrators are numerical methods that preserve geometric properties of the flow of a differential equation (see Reference [1]) This encompasses symplectic integrators for Hamiltonian systems, variational integrators for Lagrangian systems, and numerical methods on manifolds, including Lie group methods and integrators for constrained mechanical systems. By using integration by parts, the Euler–Lagrange equations follow as

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Conclusion