# CRiSP

# Collaborative Research in Structure Preservation (2011-2015)

CRiSP is a project within the International Research Staff Exchange Scheme (IRSES) - Marie Curie Actions, funded by the 7th European framework program.

Partners | Country | ||
---|---|---|---|

1 | Norwegian University of Science and Technology | Norway | Coordinator |

2 | University of Bergen | Norway | EU partner |

3 | University of Cambridge | United Kingdom | EU partner |

4 | La Trobe University (Melbourne) | Australia | Third country partner |

5 | Massey University (Palmerston North) | New Zealand | Third country partner |

#### Starting date:

**April 1st 2011**

#### Duration:

**4 years**

## People

- Elena Celledoni, NTNU, (manager of the project)
- Bynjulf Owren, NTNU
- Arieh Iserles, University of Cambridge
- Hans Munthe-Kaas, University of Bergen
- Antonella Zanna, University of Bergen
- Robert McLachlan, Massey University
- Reinout Quispel, La Trobe University

### Project summary

The goal of this project is to reinforce an existing collaboration between three European research groups and two Third country groups in the field of structure preserving numerical methods and highly oscillatory problems. The Third country partners are La Trobe University, Melbourne, Australia and Massey University, Palmerston North, New Zealand. The European beneficiaries are two Norwegian universities (NTNU, Trondheim and University of Bergen) and the University of Cambridge, UK. The main objective of our research is to develop numerical methods which exactly preserve some important geometric structure in the physical model under consideration. Typically this could mean the preservation of symplecticity in Hamiltonian systems, or the preservation of volume in divergence free systems. The research teams involved in this exchange programme have gained considerable expertise in complementary subfields of geometric numerical integration in the last two decades, and in particular Lie group methods (UiB, Cambridge, NTNU), structure preserving splitting methods (Massey, LaTrobe) and methods for highly oscillatory problems (Cambridge). The exchange will enable a transfer of knowledge between the groups including training of early stage researchers. We believe this will ultimately lead to the solution of challenging theoretical and practical problems in the structure preserving numerical solution of dynamical systems.

### Work Packages

WP nr. | WP name | Participant |
---|---|---|

1 | Energy-preserving methods | NTNU, La Trobe, Massey |

2 | Volume-preserving methods | Cambridge, UiB, La Trobe, Massey |

3 | B-series and algebraic structures | La Trobe, Massey, NTNU, UiB |

4 | Highly oscillatory problems | Cambridge, La Trobe, Massey, NTNU |

### Workshops

Worshop nr. | Workshop name | Participant | Supported by |
---|---|---|---|

1 | TRAGIC ((Tasmanian Real Analysis and Geometric Integration Conference), Launceston Tasmania, December 2010 | NTNU,UiB, Cambridge, La Trobe, Massey | Australian Academy of Science |

2 | Oberwolfach workshop, April 2011 | participants from NTNU, UiB, Cambridge, La Trobe, Massey | |

3 | Python for GI, June 2011, NTNU, Trondheim | NTNU, La Trobe, Massey | EU 7th Framework program and Norwegian Research Council |

4 | Pre-MAGIC meeting, Waikanae, New Zealand, December 4th 2011 | NTNU, UiB, La Trobe, Massey | MURST New Zealand |

5 | MAGIC, Misteltoe Bay, New Zealand, 15-19 January 2012 | La Trobe, Massey, NTNU, UiB | MURST New Zealand |

6 | Winterschule in Numerical Differential Equations, Massey University, Palmerston North, New Zealand. | NTNU, Massey | MURST New Zealand |

7 | Conference in honour of A. Iserles, Lom, Norway, September 2012 | NTNU, UiB, Cambridge, and other international participants. | Norwegian Research Council. |

8 | Algebraic Combinatorics and Numerical Analysis, December 2012 | NTNU, UiB, and other international participants. | Department of Mathematical Sciences, NTNU. |

### Secondments

Researcher | Senior/junior | Institution | Seconded to | Dates |
---|---|---|---|---|

Lina Song | ESR | La Trobe | NTNU | April 2011 |

David McLaren | SR | La Trobe | NTNU | June 2011 |

Klas Modin | ESR | Massey | NTNU | June 2011 |

Olivier Verdier | ESR | NTNU | Massey | November-December 2011 |

Tore Halvorsen | ESR | NTNU | Massey | November 2011 |

Huyen Xue | ESR | UiB | Massey | December 2011 |

Antonella Zanna | SR | UiB | Massey | December 2011 |

Hans Munthe-Kaas | SR | UiB | Massey | December 2011-January 2012 |

Morten Nome | ESR | UiB | Massey | December 2011-January 2012 |

Brynjulf Owren | SR | NTNU | Massey | January-February 2012 April-June 2012 |

Elena Celledoni | SR | NTNU | Massey | January-February 2012 April-June 2012 |

Geir Bogfjellmo | ESR | NTNU | Massey | January-February 2012 April-May 2012 |

Håkon Marthinsen | ESR | NTNU | Massey | January-February 2012 April-May 2012 |

Nataliya Ramzina | ESR | NTNU | Massey | January-February 2012 April-May 2012 |

Eirik Hoel Høiseth | ESR | NTNU | Massey | January-February 2012 April-May 2012 |

Brynjulf Owren | SR | NTNU | La Trobe | March-April 2012 |

Elena Celledoni | SR | NTNU | La Trobe | March-April 2012 |

Geir Bogfjellmo | ESR | NTNU | La Trobe | March-April 2012 |

Håkon Marthinsen | ESR | NTNU | La Trobe | March-April 2012 |

Nataliya Ramzina | ESR | NTNU | La Trobe | March-April 2012 |

Eirik Hoel Høiseth | ESR | NTNU | La Trobe | March-April 2012 |

Arieh Iserles | SR | Cambridge | Massey | March 2012 |

Arieh Iserles | SR | Cambridge | La Trobe | March 2012 |

Richard Norton | ESR | La Trobe | NTNU | August 2012 |

### Publications

Work Package | Bibliographical details |
---|---|

WP1 | Morten Dahlby, Brynjulf Owren and Takaharu Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys A 2011 |

WP1 | E. Celledoni, B. Owren Y. Sun. ``On energy preserving integrators for polynomial Hamiltonians’’, Workshop on Geometric Numerical Integration, March 20-24.Organized by: Ernst Hairer, Marlis Hochbruck, Arieh Iserles and Christian Lubich, Oberwolfach reports vol 8, issue 1, 2011. ISSN 1660-8933. |

WP1 | Morten Dahlby, Integral-preserving numerical methods for differential equations, PhD thesis, NTNU, November 2011. |

WP1 | S. Christiansen, H. Munthe-Kaas, B. Owren, Topics in structure-preserving discretization , Acta Numerica 2011. |

WP1 | E. Celledoni, R.I. McLachlan, B. Owren and G.R.W. Quispel, On conjugate B-series and their geometric structure, JNAIAAM 2011. |

WP1 | V. Grimm, R.I. McLachlan, D.I. McLaren, D.R.J. O'Neale, B. Owren, G.R.W. Quispel) Preserving energy resp. dissipation in numerical PDEs, using the "average vector field" method, arXiv:1202.4555v1. J. Comp Phys. |

WP1 | E Celledoni, B. Owren, Y Sun, The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged Vector Field method, arXiv:1203.3252v1 [math.NA], submitted. (D 1.2) |

WP2 | A. Zanna, ``Recent advancements on explicit volume-preserving splitting methods”, Workshop on Geometric Numerical Integration, March 20-24.Organized by: Ernst Hairer, Marlis Hochbruck, Arieh Iserles and Christian Lubich, Oberwolfach reports vol 8, issue 1, 2011. ISSN 1660-8933. |

WP2 | H. Xue, A. Zanna, ``Explicit volume-preserving splitting methods for polynomial divergence-free vector fields’’. BIT, Numerical Mathematics (2013). |

WP2 | A. Zanna: "The Euler equations of quasi-geostrophic fluids and volume-preserving numerical methods" Report University of Bergen. |

WP3 | Alexander Lundervold, Hans Munthe-Kaas, Backward error analysis and the substitution law for Lie group integrators, arXiv:1106.1071v1, oundations of Computational Mathematics. 13: 161-186. 2012-07-10. doi: 10.1007/s10208-012-9130-z |

WP3 | Alexander Lundervold, Hans Z. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds, arXiv:1112.4465v1 |

WP3 | Kurusch Ebrahimi-Fard, Alexander Lundervold, Simon J. A. Malham, Hans Munthe-Kaas, Anke Wiese, Algebraic structure of stochastic expansions and efficient simulation, arXiv:1112.5571v3 [math.NA]. Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. 468: 2361-2382. doi: 10.1098/rspa.2012.0024 |

WP3 | Hans Munthe-Kaas, Alexander Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames, arXiv:1203.4738v1 Foundations of Computational Mathematics. 13: 583-613. doi: 10.1007/s10208-013-9167-7 |

WP3 | Alexander Lundervold, Lie–Butcher series and geometric numerical integration on manifolds, PhD thesis, November 2011 |

WP4 | M. Condon, A. Deaño, A. Iserles & K. Kropielnicka. "Efficient computation of delay differential equations with highly oscillatory terms". NA Report Cambridge. |

WP4 | M. J. Cantero & A. Iserles, "On rapid computation of expansions in ultraspherical polynomials". NA Report Cambridge. |

WP4 | M. J. Cantero & A. Iserles, "On expansions in orthogonal polynomials". NA Report Cambridge. |

WP4 | A. Boettcher, S. Grudsky & A. Iserles. "The Fox-Li operator as a test and a spur for Wiener-Hopf theory". NA Report Cambridge. |

WP4 | M. Condon, A. Deaño & A. Iserles, "A new simulation technique for RF oscillators". NA Report Cambridge. |

WP4 | M. J. Cantero & A. Iserles, "Orthogonal polynomials on the unit circle and functional-differential equations".NA Report Cambridge. |

WP4 | M. J. Cantero & A. Iserles, "On a curious q-hypergeometric identity".NA Report Cambridge. |

WP4 | M. Condon, A. Deaño, J. Gao & A. Iserles, "Asymptotic solvers for ordinary differential equations with multiple frequencies". NA Report Cambridge. |

WP4 | S. Altinbasak, M. Condon, A. Deaño & A. Iserles, "Highly oscillatory diffusion-type equations". NA Report Cambridge. |

WP4 | A. Iserles & K. Kropielnicka "Effective approximation for linear time-dependent Schrödinger equation". NA Report Cambridge. |

WP4 | M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comp. 2011, vol 33. |

WP4 | R.I McLachlan, Some topics in multisymplectic integration, Oberwolfach Report 16/2011, pp. 14-17. DOI: 10.4171/OWR/2011/16 |

WP4 | A. Zanna, ``Generalized Polar Decompositions in Control’’. Textos de matemática : mathematical papers in honour of Fátima Silva Leite. Coimbra: Universidade de Coimbra 2011 ISBN 978-972-8564-47-6. s. 123-134. |

WP1 | E. Celledoni, B. Owren and Y. Sun, `` The minimal stage, energy-preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged vector Fiels method." arXiv:1203.3252v1. Math. Comp. |

WP1 | E. Celledoni, H. Marthinsen and B. Owren, `Àn introduction to Lie group integrators - basics, new developments and applicstions". arXiv:1207.0069. JCP. |

WP2 | A. Zanna, The Euler equation of quasi-geostrophic fluids and volume preserving numerical methods, arXiv:1205.1947v1 |

WP1/2 | E. Celledoni, R.I. McLachlan, B. Owren, GWR Quispel, Geometric properties of Kahan's method. J Phys. A |

WP1 | E. Celledoni, B. Owren, ``Preserving first integrals with symmetric Lie group methods´´, DCDS A (2014). |

WP2 | H. Xue and A. Zanna, "GENERATING FUNCTIONS AND VOLUME PRESERVING MAPPINGS", DCDS A (2014). |

WP1 | RI McLachlan and R. Quispel, "Discrete gradient methods have an energy conservation law", DCDS A (2014) |

WP3 | F. Bartha and H. Munthe-Kaas, "COMPUTING OF B-SERIES BY AUTOMATIC DIFFERENTIATION", DCDS A (2014). |

WP2 | E. Celledoni, B.K. Kometa and O. Verdier, High-order semi-Lagrangian methods for the incompressible Navier-Stokes equations, arXiv:1207.5147v1, J Sci Comp.(2015) |

WP4 | E. Celledoni, E.H. Hoiseth and N. Ramzina, Splitting methods for controlled vessel marine operations, submitted. (2014) |

WP4 | E. Celledoni, RI McLachlan, D. McLaren, B. Owren and GWR Quispel, Integrability properties of Kahan's method arXiv: 1405.3740, J. Phys A. (2014) |

WP4 | S. Blanes & A. Iserles, "Explicit adaptive symplectic integrators for solving Hamiltonian systems" Celestial Mech. & Dynamical Astronomy 114 (2012) 297–317. |

WP4 | M. Condon, A. Deaño, J. Gao & A. Iserles, "Asymptotic solvers for second-order differential equation systems with multiple frequencies" Calcolo 51 (2014), 109–139. |

WP4 | M. J. Cantero & A. Iserles, "On expansions in orthogonal polynomials", Adv. Comp. Maths 38 (2013), 35–61. |

WP4 | A. Iserles, "On skew-symmetric differentiation matrices", IMA J. Num. Anal. 34 (2014), 435–451. |

WP4 | S. Altinbasak Üsküplu, M. Condon, A. Deaño & A. Iserles, "Highly oscillatory diffusion-type equations", J. Comp. Maths. 31 (2014), 549–572. |

WP4 | M. Condon, A. Iserles & S.P. Nørsett, "Differential equations with general highly oscillatory forcing terms" , Proc. Royal Soc. A 470 (2014). |

WP4 | B. Wang & A. Iserles, "Dirichlet series for dynamical systems of first-order ordinary differential equations", Disc. & Cont. Dynamical Sys. B 19 (2014), 281–298. |

WP4 | P. Bader, A. Iserles, K. Kropielnicka & P. Singh, "Effective approximation for the linear time-dependent Schrödinger equation", to appear in Found. Comp. Maths. |

WP4 | Bin Wang, Arieh Iserles & Xinyuan Wu, "Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems", to appear in Found. Comp. Maths. |

WP4 | M. J. Cantero & A. Iserles, "From orthogonal polynomials on the unit circle to functional equations via generating functions" to appear in Transactions Amer. Maths Soc.. |

WP4 | A.G.C. P. Ramos & A. Iserles, "Numerical solution of Sturm–Liouville problems via Fer streamers" to appear in Numerische Mathematik. |

WP4 | E. Hairer & A. Iserles, "Numerical stability in the presence of variable coefficients" to appear in Found. Comp. Maths. |

WP2 | H Marthinsen and B. Owren, "Geometric integration of non-autonomous Hamiltonian problems ", arXiv:1409.5058, to appear in Num. Algs. |

WP3 | O. Verdier, H. Munthe-Kaas, Aromatic Butcher Series. Foundations of Computational Mathematics. doi: 10.1007/s10208-015-9245-0, (2015). |

WP4 | H. Munthe-Kaas, Hans, GRW Quispel, A. Zanna, 2014. Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT Numerical Mathematics. 54: 257-282. doi: 10.1007/s10543-014-0473-5. |

WP1 | RA Norton, DI McLaren, GRW Quispel, A Stern, A. Zanna, 2015. Projection methods and discrete gradient methods for preserving first integrals of ODES. Discrete and Continuous Dynamical Systems. 35: 2079-2098. |

WP4 | E. Celledoni, JM Sanz-Serna, A. Zanna, 2014. Guest editors' preface. Discrete and Continuous Dynamical Systems. 34: i-ii. doi: doi:10.3934/dcds.2014.34.3i |

WP4 | S Marsland, R McLachlan, K Modin, and M Perlmutter, On conformal variational problems and free boundary continua, J. Phys. A 47 (2014) 145204 |

WP2 | R I McLachlan, K Modin, and O Verdier, Collective symplectic integrators, Nonlinearity 27(6) (2014), 1525-1542. |

WP2 | R I McLachlan, K Modin, and O Verdier, Collective Lie-Poisson integrators on R3, IMA J. Numer. Anal. (2014). |

WP4 | R I McLachlan and A Stern, Modified trigonometric integrators, SIAM J. Numer. Anal. 52(3) (2014), 1378-1397. |

WP2 | R I McLachlan, K Modin, O Verdier, and M Wilkins, Geometric generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339-370. |

WP2 | R I McLachlan, K Modin, and O Verdier, Symplectic integrators for spin systems, Phys. Rev. E . |

WP2 | R I McLachlan, K Modin, and O Verdier, Discrete time Hamiltonian spin systems, 2014. |

WP1 | R.A. Norton and G.R.W. Quispel, ‘Discrete gradient methods for preserving a first integral of an ordinary differential equation’, Discrete and Continuous Dynamical Systems, 34 (2014), 1147-1170. |

WP3 | HS Sundklakk, A library for computing with trees and B-series, specialization project, NTNU, 2014, https://github.com/henriksu/pybs |