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
WP1E. 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.
WP1S. 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)
WP2A. 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).
WP2A. Zanna: "The Euler equations of quasi-geostrophic fluids and volume-preserving numerical methods" Report University of Bergen.
WP3Alexander 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
WP3Alexander Lundervold, Hans Z. Munthe-Kaas, On algebraic structures of numerical integration on vector spaces and manifolds, arXiv:1112.4465v1
WP3Kurusch 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
WP3Hans 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
WP4M. Condon, A. Deaño, A. Iserles & K. Kropielnicka. "Efficient computation of delay differential equations with highly oscillatory terms". NA Report Cambridge.
WP4M. J. Cantero & A. Iserles, "On rapid computation of expansions in ultraspherical polynomials". NA Report Cambridge.
WP4M. J. Cantero & A. Iserles, "On expansions in orthogonal polynomials". NA Report Cambridge.
WP4A. Boettcher, S. Grudsky & A. Iserles. "The Fox-Li operator as a test and a spur for Wiener-Hopf theory". NA Report Cambridge.
WP4M. Condon, A. Deaño & A. Iserles, "A new simulation technique for RF oscillators". NA Report Cambridge.
WP4M. J. Cantero & A. Iserles, "Orthogonal polynomials on the unit circle and functional-differential equations".NA Report Cambridge.
WP4M. J. Cantero & A. Iserles, "On a curious q-hypergeometric identity".NA Report Cambridge.
WP4M. Condon, A. Deaño, J. Gao & A. Iserles, "Asymptotic solvers for ordinary differential equations with multiple frequencies". NA Report Cambridge.
WP4S. Altinbasak, M. Condon, A. Deaño & A. Iserles, "Highly oscillatory diffusion-type equations". NA Report Cambridge.
WP4A. Iserles & K. Kropielnicka "Effective approximation for linear time-dependent Schrödinger equation". NA Report Cambridge.
WP4M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comp. 2011, vol 33.
WP4R.I McLachlan, Some topics in multisymplectic integration, Oberwolfach Report 16/2011, pp. 14-17. DOI: 10.4171/OWR/2011/16
WP4A. 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.
WP1E. 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.
WP1E. 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
2015-05-06, Elena Celledoni