R-adaptive multisymplectic and variational integrators

TL;DR

This paper introduces two novel variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories that incorporate r-adaptive mesh strategies, enhancing numerical simulation accuracy for PDEs.

Contribution

The paper develops two new r-adaptive variational integrators for field theories, integrating mesh adaptation directly into the variational framework.

Findings

Numerical results demonstrate improved accuracy for the Sine-Gordon equation.

The methods effectively couple mesh adaptation with physical field dynamics.

Advantages and limitations of each approach are discussed.

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.