### Workshop A5 - Geometric Integration and Computational Mechanics

Organizers: Fernando Casas (Universitat Jaume I, Spain) - Elena Celledoni (Norwegian University of Science and Technology, Norway) - David Martin de Diego (Instituto de Ciencias Matemáticas, Spain)

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## Talks

July 10, 14:30 ~ 15:00

## It takes a wave packet to catch a wave packet

### University of Cambridge, United Kingdom   -   ai10@cam.ac.uk

We are concerned with spectral methods for signals composed of wave packets, e.g. in quantum mechanics. The traditional approach is to use periodic boundary conditions, in which case standard Fourier methods are more than adequate, except that in long-term integration wave packets might reach the boundary and non-physical behaviour ensues. This motivates us to consider approximations on the entire real line. We consider and analyse in detail four candidates: Hermite polynomials, Hermite functions, stretched Chebyshev expansions and stretched Fourier expansions. In particular, we are concerned with the speed of convergence of standard and $m$-term approximations. And the winner is…

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July 10, 15:00 ~ 15:30

## Uniformly accurate averaging numerical schemes for oscillatory evolution equations.

### INRIA, France   -   philippe.chartier@inria.fr

In this talk, we consider diff erential equations with (possibly high) oscillations whose frequency scales like the inverse of a parameter with values in an interval of the form ]0; 1]. In particular, we do not envisage the (possibly highly-) oscillating system in its asymptotic regime. Our aim is to design a numerical method which can solve such systems at a cost and with an accuracy remaining essentially insensitive to the value of the parameter.

Joint work with Mohammed Lemou, Florian Mehats and Gilles Vilmart.

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July 10, 15:30 ~ 16:00

## Magnus-type integrators combined with operator splitting methods and their areas of applications

### Leopold--Franzens-Universität Innsbruck, Austria   -   Mechthild.Thalhammer@uibk.ac.at

In this talk, I shall introduce the class of commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations and identify different areas of application.

Commutator-free quasi-Magnus exponential integrators are (formally) given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Avoiding the evaluation of commutators, they provide a favourable alternative to standard Magnus integrators.

Non-autonomous linear evolution equations also arise as a part of more complex problems, for instance in connection with nonlinear evolution equations of the form $u'(t) = A(t) u(t) + B(u(t))$. A natural approach is thus to apply commutator-free quasi-Magnus exponential integrators combined with operator splitting methods. Relevant applications include Schrödinger equations with space-time-dependent potential describing Bose-Einstein condensation or diffusion-reaction systems modelling pattern formation.

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July 10, 16:00 ~ 16:30

## Energy preserving time integrators for PDEs on moving grids

### NTNU, Norway   -   brynjulf.owren@ntnu.no

Energy preserving, or more generally, integral preserving schemes for ODEs can be derived by means of for instance discrete gradient methods. For PDEs, one may first discretize in space, using for instance finite difference methods or finite element methods, and then apply an integral preserving method for the corresponding ODEs. For PDEs discretized on moving grids, the situation is more complicated, it is not even clear exactly what should be meant by an integral preserving scheme in this setting. We shall propose a definition and then derive the resulting conservative schemes, both with finite difference schemes and with finite element schemes. We test the methods on problems with travelling wave solutions and demonstrate that they give remarkably good results, both compared to fixed grid and to non-conservative schemes

Joint work with Sølve Eidnes (NTNU) and Torbjørn Ringholm (NTNU).

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July 10, 17:00 ~ 17:30

## The spherical midpoint method

### Chalmers and University of Gothenburg, Sweden   -   klas.modin@chalmers.se

The 2-sphere $S^2$ is, perhaps, the most fundamental example of a non-canonical symplectic manifold. Yet, to construct symplectic integration schemes for Hamiltonian systems on $S^2$ has been surprisingly cumbersome. In this talk I shall present a new integrator---the spherical midpoint method---for general Hamiltonian systems on $S^2$. The new method uses a minimal number of variables, is equivariant with respect to the homogeneous space structure of the 2-sphere, and is readily extendable to general Hamiltonian systems on $(S^2)^m\times \mathbb{R}^{2n}$. I shall also discuss applications to atomistic spin dynamics in condensed matter physics (collaboration with physicists at Uppsala University), and a possible generalization of the method to other Kähler manifolds.

Joint work with Robert McLachlan (Massey University, New Zealand) and Olivier Verdier (Western Norway University of Applied Sciences).

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July 10, 17:30 ~ 18:00

## Higher order variational integrators and their relation to Runge-Kutta methods for unconstrained and constrained systems

### University of Oxford, United Kingdom   -   sina.ober-blobaum@eng.ox.ac.uk

In this talk higher order variational integrators are developed and their relation to Runge-Kutta methods is investigated. While it is well known that particular higher order variational integrators are equivalent to symplectic partitioned Runge- Kutta methods, this is not true for more generally constructed variational integrators. Based on existing results, modified schemes of Runge-Kutta integrators are derived and shown to be equivalent to a new class of higher order variational integrators which distinguishes in the dimension of the function space for approximating the solution curves compared to classical Galerkin variational integrators. Conditions are derived under which both variational integrators are identical and demonstrated numerically by simple mechanical examples. The results are extended to systems with holonomic constraints.

Joint work with Sigrid Leyendecker (University of Erlangen-Nuremberg, Germany), Theresa Wenger (University of Erlangen-Nuremberg, Germany).

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July 10, 18:00 ~ 18:30

## The inverse problem for discrete systems

### ICMAT, Spain   -   marta.farre@icmat.es

The classical inverse problem of the calculus of variations consists in determining whether or not a given system of explicit second order differential equations is equivalent to a system of Euler-Lagrange equations for some regular Lagrangian. We will develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics, using suitable Lagrangian and isotropic submanifolds. We will also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design of geometric integrators.

Joint work with María Barbero Liñán (Universidad Politécnica de Madrid, Spain), Sebastián Ferraro (Universidad Nacional del Sur, Argentina) and David Martín de Diego (ICMAT, Spain).

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July 11, 14:30 ~ 15:20 - Room 111

## The Clebsch representation in optimal control, integrable systems and discrete dynamics

### University of Michigan , USA   -   abloch@umich.edu

In this talk we discuss certain kinematic optimal control problems (the Clebsch problems) and their connection to classical integrable systems. In particular, we consider the rigid body problem and its low rank counterparts, the geodesic flows on Stiefel manifolds and their connection with the work of Moser, flows on symmetric matrices, and the Toda flows. We also discuss discrete formulations of these systems and their connection with numerical algorithms.

Joint work with Francois Gay-Balmaz (Ecole Normale Superieure, Paris) and Tudor Ratiu (Shanghai Jiao Tong University and EPFL (Shanghai and Lausanne).

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July 11, 15:30 ~ 16:00 - Room 111

## How do nonholonomic integrators work?

### Western Norway University of Applied Sciences, Norway   -   olivier.verdier@hvl.no

Nonholonomic systems are mechanical systems with constraints on the velocity. Their behaviour is quite different from that of mechanical systems with constraints on the positions (holonomic systems). There has been reports in the literature of integrators which behaved particularly well for some nonholonomic systems: near conservation of energy, or near conservation of other integrals.

We will explain the general mechanism behind those good properties. The main structure of the examples where the nonholonomic integrators work is that of a fibration over a reversible integrable system. The explaining theory, for descending integrators, is then the reversible Kolmogorov–Arnold–Moser theory.

We will explain how to design various systems which deviate from that pattern, in order to show experimentally that this structure of fibration over a reversible integrable system is necessary. Non-holonomic integrators do not preserve any of the features of those perturbed systems: neither the energy, nor the integrable structure.

We will also single out one non-holonomic integrator, which has two properties of interests: it is semi-implicit for a large range of systems, and it still has a good, completely unexplained, behaviour on some nonholonomic systems.

Joint work with Klas Modin (Chalmers University of Technology).

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July 11, 16:00 ~ 16:30 - Room 111

## On unit-quaternion based Galerkin Lie group variational integrators

### University of Erlangen-Nuremberg, Germany   -   sigrid.leyendecker@fau.de

Lie-group variational integrators are often used to simulate rigidy body dynamics~\cite{Leitz2014,Terze2016}. Using unit quaternions is an efficient way to represent the rotational degrees of freedom of a rigid body~\cite{Betsch2009}. This is in part due to the fact, that they facilitate a simple interpolation method presented in this talk. This interpolation method, toghether with an appropriate quadrature rule, is then used to approximate the action of the dynamical system with rotational degrees of freedom and a Lie-group variational integrator of arbitrary order is derived.

We present a numerical convergence analysis, both on the main grid, as well as for the whole Galerkin curve. Our convergence analysis for the whole Galerkin curve is in agreement with the results of Hall et al.~\cite{Hall2015,Hall2015a}, but we also investigate the convergence on the main grid, where the convergence rate is considerably higher. Furthermore, we use unit quternions instead of rotation matrices, which might be computationally more efficient. Our interpolation method does not require a change of coordinates in the momentum matching part of the discrete Euler-Lagrange equations as the one used in~\cite{Hall2015}. The computational efficiency is investigated by showing the relationship between the error and the CPU-time.

The same method can be extended to derive multisymplectic Galerkin Lie-group variational integrators, e.g.~for the simulation of geometrically exact beam dynamics, by applying the interpolation method to the two-dimensional space time domain.

\begin{thebibliography}{1}

\bibitem{Betsch2009} Peter Betsch and Ralf Siebert. \newblock Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration. \newblock {\em International Journal for Numerical Methods in Engineering}, 79(4):444--473, 2009.

\bibitem{Hall2015} James Hall and Melvin Leok. \newblock Lie group spectral variational integrators. \newblock {\em Foundations of Computational Mathematics}, pages 1--59, 2015.

\bibitem{Hall2015a} James Hall and Melvin Leok. \newblock Spectral variational integrators. \newblock {\em Numerische Mathematik}, 130(4):681--740, 2015.

\bibitem{Leitz2014} Thomas Leitz, Sina Ober-Bl{\"{o}}baum, and Sigrid Leyendecker. \newblock Variational integrators for dynamical systems with rotational degrees of freedom. \newblock In {\em Proceedings of WCCM XI - ECCM V - ECFD VI}, pages 3148--3159, 2014.

\bibitem{Terze2016} Zdravko Terze, Andreas M{\"{u}}ller, and Dario Zlatar. \newblock Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations. \newblock {\em Multibody System Dynamics}, pages 1--25, 2016.

\end{thebibliography}

Joint work with Thomas Leitz (University of Erlangen-Nuremberg).

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July 11, 17:00 ~ 17:30 - Room 111

## ROUGH PATHS ON HOMOGENEOUS SPACES

### Bergen, Norway   -   hans.munthe-kaas@uib.no

We consider rough paths in the context of differential equations on homogenous manifolds. Our approach generalises both the Chen-type shuffle algebra of word series underlying Terry Lyons’ theory of rough paths as well as the branched rough paths of Massimiliano Gubinelli, which are based on B-series and the Connes-Kreimer Hopf algebra of rooted trees. The new approach contains these as special cases and extends geometrically to rough paths evolving on homogeneous manifolds.

References:

[1] K. T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math. 65 No1, 163–178 (1957).

[2] M. Gubinelli, Ramification of Rough Paths, Journal of Differential Equations 248, no. 4 (2010) 693-721.

[3] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 No2, 215–310 (1998).

Joint work with Charles Curry (NTNU, Norway), Kurusch Ebrahimi-Fard (NTNU, Norway) and Dominique Manchon (Clermont-Ferrand, France).

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July 11, 17:30 ~ 18:00 - Room 111

## Action algebroids are the only post-Lie algebroids

### Washington University in St. Louis, USA   -   stern@wustl.edu

Despite their name, Lie group integrators may also be applied to certain manifolds that are not Lie groups: homogeneous spaces, for example. In general, Munthe-Kaas and Lundervold showed that Lie-Butcher series methods may be defined on vector bundles called post-Lie algebroids. We show that every post-Lie algebroid arises from a Lie algebra action, meaning that manifolds with Lie algebra actions are essentially the most general spaces to which Lie-Butcher series methods may be applied.

Joint work with Hans Z. Munthe-Kaas (University of Bergen) and Olivier Verdier (Bergen University College).

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July 11, 18:00 ~ 18:30 - Room 111

## Palindromic 3-stage splitting integrators, a roadmap

### Yachay Tech, Ecuador   -   cedricmc@yachaytech.edu.ec

The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.

Joint work with J.M. Sanz-Serna.

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July 11, 18:30 ~ 19:00 - Room 111

## Analyzing splitting stochastic integrators using word series

### Alfonso Álamo Alfonso Álamo Zapatero

Recently, word series expansion have been suggested as an alternative to B-series. Word series are similar to, but simpler than, the B-series used to analyze Runge-Kutta and other one-step integrators, although its scope of applicability is narrower than that of B-series. We shall show how to extend them to the stochastic case in which word series turn out to be again an extremely useful tool for the analysis of split-step integradors, yielding a systematic way to obtain, for example, modified equations and order conditions.

Joint work with J.M. Sanz-Serna (Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain. Email: jmsanzserna@gmail.com).

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July 12, 14:30 ~ 15:20 - Room 111

## Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics

### Imperial college London, UK   -   d.holm@imperial.ac.uk

In [Holm, Proc. Roy. Soc. A 471 (2015)] stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby justifying stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centering condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

Joint work with Colin J Cotter (Imperial college London) Georg A Gottwald (University of Sydney).

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July 12, 15:30 ~ 16:00 - Room 111

## Variational integrators in geophysical fluid dynamics

### CNRS & Ecole Normale Supérieure de Paris, France   -   gaybalma@lmd.ens.fr

We present an overview of structure-preserving variational discretizations of various equations of geophysical fluid dynamics, such as the anelastic, pseudo-incompressible, and shallow-water equations. These discretizations rely on finite dimensional approximations of the groups of diffeomorphisms underlying the dynamics. In particular, we extend previous approches to the compressible case. As descending from variational principles, the discussed variational schemes exhibit discrete versions of Kelvin circulation theorems, are applicable to irregular meshes, and show excellent long term energy behavior.

Joint work with Werner Bauer (Imperial College, London).

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July 12, 16:00 ~ 16:30 - Room 111

## Interpolation of Manifold-Valued Functions via the Generalized Polar Decomposition

### University of California, San Diego, United States   -   egawlik@ucsd.edu

We construct interpolation operators for manifold-valued functions, with an emphasis on functions taking values in symmetric spaces and Lie groups. A key role in our construction is played by the polar decomposition -- the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix -- and its generalization to Lie groups. We demonstrate that this factorization can be leveraged to carry out a number of seemingly disparate tasks, including the design of finite elements for numerical relativity, the interpolation of subspaces for reduced-order modeling, and the approximation of Riemannian cubics on the special orthogonal group.

Joint work with Melvin Leok (University of California, San Diego).

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July 12, 17:00 ~ 17:30 - Room 111

## Commutator-free Magnus based methods

### Polish Academy of Sciences, Poland   -   karolina.kropielnicka@mat.ug.edu.pl

In this talk, I shall introduce the class of commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations and identify different areas of application.

Commutator-free quasi-Magnus exponential integrators are (formally) given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Avoiding the evaluation of commutators, they provide a favourable alternative to standard Magnus integrators.

Non-autonomous linear evolution equations also arise as a part of more complex problems, for instance in connection with nonlinear evolution equations of the form u'(t) = A(t) u(t) + B(u(t)). A natural approach is thus to apply commutator-free quasi-Magnus exponential integrators combined with operator splitting methods. Relevant applications include Schrödinger equations with space-time-dependent potential describing Bose-Einstein condensation or diffusion-reaction systems modelling pattern formation

Joint work with Philipp Bader (La Trobe University, Australia), Iserles Arieh (University of Cambridge, UK), Pranav Singh (University of Oxford, UK).

View abstract PDF

July 12, 17:30 ~ 18:00 - Room 111

## Geometric integrators and optimal control for real-time control of robotic systems

### Johns Hopkins University, United States   -   marin@jhu.edu

This talk will focus on geometric integrators and trajectory optimization of physical systems consisting of rigid bodies and actuators such as propellers, joint motors, or wheels. Since the integrators in general are implicit, the time-step is chosen in order to guarantee convergence. Therefore, even with a large time step one could preserve properties such as momentum conservation while still being able to solve the system dynamics correctly. These integrators are then used for optimal control, using a projected stage-wise Newton method which using standard regularization and line-search techniques provides a locally optimal solution. The methods are applied to robotic systems including aerial and ground vehicles, resulting in millisecond run-times suitable for on-board implementation.

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July 12, 18:00 ~ 18:30 - Room 111

## An improved algorithm to compute the exponential of a matrix

### Universidad Politécnica de Valencia, Spain   -   serblaza@imm.upv.es

Exponential integrators have shown to be highly efficient geometric integrators, in particular as Lie group methods. However, for matrices of moderate size, their performance depends on an efficient computation of the matrix exponential. This can be achieved by the scaling-squaring technique with an appropriate approximation to the Taylor expansion of the exponential. Pad\’e approximants allows to approximate polynomial matrix functions with relatively few products of matrices (the cost to compute the inverse of a matrix is 4/3 the cost of one product). Recently, the Paterson-Stockmeyer scheme has shown to be superior to Pad\’e in many cases, since only requires $2(k-1)$ products to compute a polynomial of degree $k2$.

Taking into account that with $k$ products one can build matrix polynomials of degree $2^k$, we consider the inverse problem. Given a matrix polynomial, we analyse how to factorize it with the minimum number of products. For example, a polynomial of degree 8 can be obtained with only 3 products. We consider the decomposition of higher order polynomials and its application to approximate the exponential of a matrix and its performance with respect to the algorithm “expm” used in Matlab.

Joint work with Fernando Casas and Philipp Bader.

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## Hamel's Formalism for Infinite-Dimensional Mechanical Systems

### Washington University in St. Louis, United States   -   yasha@wustl.edu

This poster presents results of a paper of the same title. We introduce Hamel’s formalism for infinite-dimensional mechanical systems and in particular consider its applications to the dynamics of nonholonomically constrained systems. This development is a nontrivial extension of its finite-dimensional counterpart. The analysis is applied to several continuum mechanical systems of interest, including coupled systems and systems with infinitely many constraints.

Joint work with Donghua Shi (Beijing Institute of Technology), Dmitry V. Zenkov (North Carolina State University) and Anthony M. Bloch (University of Michigan).

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## Time-average Magnus-decomposition methods for solving non-autonomous linear wave equations

### Universitat Politècnica de València, Spain   -   nikop1@upvnet.upv.es

The work presented addresses the numerical integration of the second-order time-dependent linear partial differential equation $u_{tt}(x,t)= f(t,x)\, u(x,t),\qquad x\in \mathbb{R}^d,\ t\geq0,$ equipped with the initial conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u'_0(x)$.

After spatial discretization, the equation can be expressed as $y''(t) = M(t) y(t) , \quad y(t_0)=y_0,\ y'(t_0)=y_0',$ where $t \in \mathbb{R}$, $y \in \mathbb{C}^{r}$.

In general, the solution is oscillatory, and exponential integrators show a good performance since they, in turn, provide numerical solutions which reflect the oscillatory nature of the exact solution.

However, due to the large dimension of the problem, one has consider thoroughly the computational cost of the schemes. As $M(t)$ originates from the discretization of a PDE, methods that employ only matrix--vector products are computationally reasonable. Thus, one of the promising approaches is to approximate exponentials in Magnus-decomposition methods by employing Krylov-type subspace methods.

We compared the solutions of non-autonomous problems obtained by the new methods, by their predecessors and by a set of well-known methods. The results show that Krylov-type exponential computation facilitates application of Magnus-decomposition methods, which were initially designed for solving matrix differential equations, to large-dimensional problems.

Joint work with Philipp Bader (Universitat Jaume I), Sergio Blanes (Universitat Politècnica de València) and Fernando Casas (Universitat Jaume I).

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## Higher-order Geometric Nonholonomic Integrators on Vector Spaces and Lie Groups

### ICMAT, Spain   -   rt.sato@ucm.es

We have obtained geometrically consistent arbitrarily high-order partitioned Runge-Kutta integrators for nonholonomic systems both on vector spaces and Lie groups. These methods can be understood as a natural extension of symplectic and variational integrators in the realm of nonholonomic systems and fit nicely into a nonholonomic Hamilton-Jacobi framework. These methods differ from those of J. Cortés and S. Martínez in that we do not require the discretisation of the constraint. Our methods preserve the continuous constraint exactly and can be seen to extend those of M. de León, D. Martín de Diego and A. Santamaría.

Joint work with David Martín de Diego, ICMAT.

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## Stucture Preserving Integration of Non-spherical Particles in Turbulent Flows

### NTNU, Norway   -   benjamin.tapley@ntnu.no

The dynamics of point, non-spherical particles in turbulent flows are considered. The flows are simulated with a direct numerical simulation (DNS) strategy and the particles are modeled as rigid bodies with torques calculated from the Jeffrey equations [1]. The particle dimensions are assumed to be lower than the Kolmogorov scale, the minimum length scale at which vortices remain stable. Typical large-scale computations involve systems with many particles (e.g., 10^6), whose trajectories are obtained by integrating in time 13 ordinary differential equations per particle [2]; hence, computational cost becomes a significant factor when deciding on an appropriate numerical scheme. Implementing appropriate geometric numerical integration techniques can improve the performance of the simulation codes thus allowing for larger time-steps in the integration of the particle trajectories whilst maintaining similar levels of accuracy when compared to conventional integrators. In this poster, we implement structure preserving splitting methods that use Jacobi elliptic functions on the free rigid body equations to model the particle torques. This solution is compared against conventional integrators (e.g., Runge-Kutta or multi-step type integrators) and a reference solution. Such a method will lead to more stable and accurate numerical solutions when compared to conventional methods, and therefore reduce computational cost.

References:

[1] G. B. Jeffery, The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 102, No. 715 (Nov. 1, 1922), pp. 161-179

[2] Celledoni, E., Fasso, F., Säfström, N. and Zanna, A., The exact computation of the free rigid body motion and its use in splitting methods. SIAM Journal on Scientific Computing, 2008. 30(4): p. 2084-2112

Joint work with Elena Celledoni (NTNU, Norway).

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## Mixed order variational integrators for multiscale problems

### University of Erlangen-Nuremberg, Germany   -   theresa.wenger@fau.de

The numerical treatment of mechanical systems showing dynamics on different time scales, caused e.g. by different types or stiffnesses in potentials, is challenging. The required accuracy for a stable integration is dictated by the fast motion, leading to unacceptable high computational costs. To construct a more efficient integrator, the idea here is to separate the unknown configurations $q$ into fast and slow degrees of freedom and use different polynomials to approximate each. Furthermore, the contributions of the action are approximated via quadrature rules of different order. Applying Hamilton's principle provides the variationally derived integration scheme. If the system is holonomically constrained, the action is augmented by the integral of $g(q) \cdot \lambda$, with $\lambda$ being the Lagrange-multiplier. The constraints $g$ are split into a part depending only on slow degrees of freedom and another part coupling the whole configuration. This distinction enables different approximations, which are subject to some restrictions that ensure the solvability of the discrete equations and prevent a drift off the constraint manifold. The properties of the presented integrators of mixed order are considered such as energy and momentum maps preservation, time reversibility and stability.

Joint work with Sigrid Leyendecker (University of Erlangen-Nuremberg, Germany) and Sina Ober-Blöbaum (University of Oxford, UK).

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## Lie group integrators commute up to order four

### University of Kent, UK   -   mz233@kent.ac.uk

Let $\mathcal{M}$ be a manifold, $\mathcal{G}$ a Lie group and $\mathfrak{g}$ its associated Lie algebra. We wish to integrate systems of the form $\begin{cases} \rho_x = Q^x\rho \\ \rho_y=Q^y\rho \end{cases}$ where $\rho: \mathcal{M} \rightarrow \mathcal{G}$ is a moving frame and $Q^x,Q^y: \mathcal{M} \rightarrow \mathfrak{g}$. The compatibility condition for the system above can be expressed as $R(Q^x,Q^y):= \frac{\partial}{\partial y} Q^x - \frac{\partial}{\partial x} Q^y - [Q^y,Q^x]=0$ We investigate whether the Lie group integrators discussed in [2] commute when applied in the two different directions. Specifically we seek to show that if the integrals for $\rho$ are calculated along the two paths represented by $(x_0,y_0) \rightarrow (x_0+h,y_0) \rightarrow (x_0+h, y_0+k)$ and $(x_0,y_0) \rightarrow (x_0,y_0+k) \rightarrow (x_0+h, y_0+k),$ where $h,k$ are the stepsizes in the two directions of a discretised region of $\mathbb{R}^2$, then the results are the same.

To this end, we write $Q^x$ and $Q^y$ as symbolic Taylor expansions around the point $(x_0,y_0)$ and insert those into the relevant Magnus and BCH expansions. We show the integrators commute up to order 4 in the sense that the coefficients of $h^ik^j \, \left(i+j\leq 4\right)$ are linear differential expressions of $R(Q^x,Q^y)$ and are hence zero. The commutation is a testament to the geometry built into the Lie group integrators [2].

Our application is to the study of Lie group invariant variational systems, where the moving frame, $\rho$, plays a pivotal role [1,3].

References:

[1] T. M. N. Goncalves and E. L. Mansfield, On Moving Frames and Noether’s conservation laws, 2012, Studies in Applied Mathematics, Volume 128, Issue 1

[2] A. Iserles, H.Z. Munthe-Kaas, S.P. Norsett, A. Zanna, Lie-group methods, 2005, Acta Numerica

[3] E.L. Mansfield, A Practical Guide to the Invariant Calculus, 2010, Cambridge University Press, Cambridge

Joint work with Elizabeth Mansfield (University of Kent, UK).

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FoCM 2017, based on a nodethirtythree design.