# R-Adaptive Multisymplectic and Variational Integrators

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## Abstract

**:**

## 1. Introduction

#### 1.1. Overview

- Discretization of the physical PDE
- Mesh adaptation strategy
- Coupling the mesh equations to the physical equations

#### 1.2. Outline

## 2. Control-Theoretic Approach to r-Adaptation

#### 2.1. Reparametrized Lagrangian

**Proposition**

**1.**

**Proof.**

#### 2.2. Spatial Finite Element Discretization

#### 2.3. Differential-Algebraic Formulation and Time Integration

**Proposition**

**2.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

#### 2.4. Moving Mesh Partial Differential Equations

#### 2.5. Example

#### 2.6. Backward Error Analysis

## 3. Lagrange Multiplier Approach to r-Adaptation

#### 3.1. Reparametrized Lagrangian

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**1.**

#### 3.2. Spatial Finite Element Discretization

#### 3.3. Invertibility of the Legendre Transform

**Proposition**

**7.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

#### 3.4. Existence and Uniqueness of Solutions

**Proposition**

**8.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.5. Constraints and Adaptation Strategy

#### 3.5.1. Global Constraint

**Proposition**

**9.**

**Proof.**

#### 3.5.2. Local Constraint

#### 3.6. DAE Formulation of the Equations of Motion

**Proposition**

**10.**

**Proof.**

#### 3.7. Backward error analysis

## 4. Multisymplectic Field Theory Formalism

#### 4.1. Background Material

#### 4.1.1. Lagrangian Mechanics and Veselov-Type Discretizations

#### 4.1.2. Multisymplectic Geometry and Lagrangian Field Theory

#### 4.1.3. Multisymplectic Variational Integrators

#### 4.2. Analysis of the Control-Theoretic Approach

#### 4.2.1. Continuous Setting

#### 4.2.2. Discretization

#### 4.3. Analysis of the Lagrange Multiplier Approach

#### 4.3.1. Continuous Setting

**Theorem**

**1**

**(Lagrange**

**multiplier**

**theorem).**

- 1.
- $\sigma \in \mathcal{N}$ is an extremum of ${h|}_{\mathcal{N}}$,
- 2.
- There exists an extremum $\overline{\sigma}\in \mathcal{E}$ of $\overline{h}:\mathcal{E}\u27f6\mathbb{R}$ such that ${\pi}_{\mathcal{M},\mathcal{E}}(\overline{\sigma})=\sigma $,

#### 4.3.2. Discretization

**Remark**

**5.**

## 5. Numerical Results

#### 5.1. The Sine–Gordon Equation

#### 5.2. Generating Consistent Initial Conditions

#### 5.3. Convergence

#### 5.4. Energy Conservation

## 6. Summary and Future Work

#### 6.1. Non-Hyperbolic Equations

#### 6.2. Hamiltonian Field Theories

#### 6.3. Time Adaptation Based on Local Error Estimates

#### 6.4. Constrained Multisymplectic Field Theories

#### 6.5. Mesh Smoothing and Variational Nonholonomic Integrators

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) If ${\gamma}_{k-1}\ne {\gamma}_{k}$, then any change to the middle point changes the local shape of $\varphi (X,t)$. (

**Right**) If ${\gamma}_{k-1}={\gamma}_{k}$, then there are infinitely many possible positions for $({X}_{k},{y}_{k})$ that reproduce the local linear shape of $\varphi (X,t)$.

**Figure 4.**The single soliton solution obtained with the Lagrange multiplier strategy for $N=15$. Integration in time was performed using the 4th order Lobatto IIIA–IIIB scheme for constrained mechanical systems. The soliton moves to the right with the initial velocity $v=0.9$, bounces from the right wall at $t=13.84$, and starts moving to the left with the velocity $v=-0.9$, towards the left wall, from which it bounces at $t=41.52$.

**Figure 5.**The single soliton solution obtained with the Lagrange multiplier strategy for $N=31$: Integration in time was performed using the 4th-order Lobatto IIIA–IIIB scheme for constrained mechanical systems.

**Figure 6.**The single soliton solution obtained with the control-theoretic strategy for $N=22$: Integration in time was performed using the 4th-order Gauss scheme. Integration with the 4th-order Lobatto IIIA-IIIB yields a very similar level of accuracy.

**Figure 7.**The single soliton solution obtained with the control-theoretic strategy for $N=31$: Integration in time was performed using the 4th-order Gauss scheme. Integration with the 4th-order Lobatto IIIA–IIIB yields a very similar level of accuracy.

**Figure 8.**The mesh point trajectories (with zoomed-in insets) for the Lagrange multiplier strategy for $N=22$ (

**left**) and $N=31$ (

**right**): Integration in time was performed using the 4th-order Lobatto IIIA–IIIB scheme for constrained mechanical systems.

**Figure 9.**The mesh point trajectories (with zoomed-in insets) for the control-theoretic strategy for $N=22$ (

**left**) and $N=31$ (

**right**): Integration in time was performed using the 4th-order Gauss scheme. Integration with the 4th-order Lobatto IIIA–IIIB yields a very similar result.

**Figure 10.**The single soliton solution computed on a uniform mesh with $N=31$: Integration in time was performed using the 4th-order Gauss scheme. Integration with the 4th-order Lobatto IIIA–IIIB yields a very similar level of accuracy.

**Figure 11.**Comparison of the convergence rates of the discussed methods: Integration in time was performed using the 4th-order Lobatto IIIA–IIIB method for constrained systems in case of the Lagrange multiplier strategy and the 4th-order Gauss scheme in case of both the control-theoretic strategy and the uniform mesh simulation. The 4th-order Lobatto IIIA–IIIB scheme for the control-theoretic strategy and the uniform mesh simulation yield a very similar level of accuracy. Also, using 2nd-order integrators gives very similar error plots.

**Figure 12.**The two-soliton solution obtained with the control-theoretic and Lagrange multiplier strategies for $N=25$. Integration in time was performed using the 4th-order Gauss quadrature for the control-theoretic approach and the 4th-order Lobatto IIIA–IIIB quadrature for constrained mechanical systems in case of the Lagrange multiplier approach. The solitons initially move towards each other with the velocities $v=0.9$, then bounce off of each other at $t=5$ and start moving towards the walls, from which they bounce at $t=18.79$. The solitons bounce off of each other again at $t=32.57$. This solution is periodic in time with the period ${T}_{period}=27.57$. The nearly exact solution was constructed in a similar fashion as Equation (168). As the simulation progresses, the Lagrange multiplier solution gets ahead of the exact solution, whereas the control-theoretic solution lags behind.

**Figure 13.**The discrete energy ${E}_{N}$ for the Lagrange multiplier strategy: Integration in time was performed with the 2nd (

**top**) and 4th (

**bottom**) order Lobatto IIIA–IIIB method for constrained mechanical systems. The spikes correspond to the times when the solitons bounce off of each other or of the walls.

**Figure 14.**The discrete energy ${E}_{N}$ for the control-theoretic strategy: Integration in time was performed with the 4th-order Gauss (

**top**), 4th-order Lobatto IIIA–IIIB (

**middle**), and non-symplectic 5th-order Radau IIA (

**bottom**) methods.

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Tyranowski, T.M.; Desbrun, M.
R-Adaptive Multisymplectic and Variational Integrators. *Mathematics* **2019**, *7*, 642.
https://doi.org/10.3390/math7070642

**AMA Style**

Tyranowski TM, Desbrun M.
R-Adaptive Multisymplectic and Variational Integrators. *Mathematics*. 2019; 7(7):642.
https://doi.org/10.3390/math7070642

**Chicago/Turabian Style**

Tyranowski, Tomasz M., and Mathieu Desbrun.
2019. "R-Adaptive Multisymplectic and Variational Integrators" *Mathematics* 7, no. 7: 642.
https://doi.org/10.3390/math7070642