Advertisement

Why Geometric Numerical Integration?

Conference paper
  • 406 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

Abstract

Geometric numerical integration (GNI) is a relatively recent discipline, concerned with the computation of differential equations while retaining their geometric and structural features exactly. In this paper we review the rationale for GNI and review a broad range of its themes: from symplectic integration to Lie-group methods, conservation of volume and preservation of energy and first integrals. We expand further on four recent activities in GNI: highly oscillatory Hamiltonian systems, W. Kahan’s ‘unconventional’ method, applications of GNI to celestial mechanics and the solution of dispersive equations of quantum mechanics by symmetric Zassenhaus splittings. This brief survey concludes with three themes in which GNI joined forces with other disciplines to shed light on the mathematical universe: abstract algebraic approaches to numerical methods for differential equations, highly oscillatory quadrature and preservation of structure in linear algebra computations.

Keywords

Symplectic methods Lie-group methods Splittings Exponential integrators Kahan’s method Variational integrators Preservation of integrals Preservation of volume 

MSC numbers

Primary 65L05 Secondary 65P10 65M99 17B45 57S50 

Notes

Acknowledgements

This work has been supported by the Australian Research Council. The authors are grateful to David McLaren for assistance during the preparation of this paper, as well as to Philipp Bader, Robert McLachlan and Marcus Webb, whose comments helped to improve this paper.

References

  1. 1.
    Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Effective approximation for the semiclassical Schrödinger equation. Found. Comput. Math. 14(4), 689–720 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. Proc. R. Soc. A 472(2193), 20150733 (2016)CrossRefGoogle Scholar
  3. 3.
    Benner, P., Fassbender, H., Stoll, M.: Solving large-scale quadratic eigenvalue problems with Hamiltonian eigenstructure using a structure-preserving Krylov subspace method. Electron. Trans. Numer. Anal. 29, 212–229 (2007/08)Google Scholar
  4. 4.
    Blanes, S., Casas, F., Murua, A.: Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Mat. Apl. 45, 89–145 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl. Numer. Math. 68, 58–72 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470(5–6), 151–238 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Butcher, J.C.: Coefficients for the study of Runge-Kutta integration processes. J. Austral. Math. Soc. 3, 185–201 (1963)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Celledoni, E., Iserles, A.: Methods for the approximation of the matrix exponential in a Lie-algebraic setting. IMA J. Numer. Anal. 21(2), 463–488 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “average vector field” method. J. Comput. Phys. 231(20), 6770–6789 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Integrability properties of Kahan’s method. J. Phys. A 47(36), 365202, 20 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Discretization of polynomial vector fields by polarization. Proc. R. Soc. A 471, 20150390 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Geometric properties of Kahan’s method. J. Phys. A 46(2), 025201, 12 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chartier, P., Murua, A.: Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27(2), 381–405 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cohen, D., Gauckler, L., Hairer, E., Lubich, C.: Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions. BIT 55(3), 705–732 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Connes, A., Kreimer, D.: Lessons from quantum field theory: Hopf algebras and spacetime geometries. Lett. Math. Phys. 48(1), 85–96 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Deaño, A., Huybrechs, D., Iserles, A.: Computing Highly Oscillatory Integrals, SIAM (2018)Google Scholar
  17. 17.
    Duncan, M., Levison, H.F., Lee, M.H.: A multiple time step symplectic algorithm for integrating close encounters. Astron. J. 116, 2067–2077 (1998)CrossRefGoogle Scholar
  18. 18.
    Ebrahimi-Fard, K., Manchon, D.: A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. 9(3), 295–316 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Farrés, A., Laskar, J., Blanes, S., Casas, F., Makazaga, J., Murua, A.: High precision symplectic integrators for the Solar System. Celest. Mech. Dyn. Astron. 116(2), 141–174 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Feng, K., Shang, Z.J.: Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71(4), 451–463 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Feng, K., Wu, H.M., Qin, M.Z., Wang, D.L.: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math. 7(1), 71–96 (1989)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Fer, F.: Résolution de l’équation matricielle \(dU/dt=pU\) par produit infini d’exponentielles matricielles. Acad. Roy. Belg. Bull. Cl. Sci. 5(44), 818–829 (1958)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Forest, É.: Geometric integration for particle accelerators. J. Phys. A 39(19), 5321–5377 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gauckler, L., Hairer, E., Lubich, C.: Energy separation in oscillatory Hamiltonian systems without any non-resonance condition. Commun. Math. Phys. 321(3), 803–815 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ge, Z., Marsden, J.E.: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133(3), 134–139 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Grimm, V., McLachlan, R.I., McLaren, D.I., Quispel, G.R.W., Schönlieb, C.-B.: Discrete gradient methods for solving variational image regularization models. J. Phys. A 50, 295201 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hairer, E.: Energy-preserving variant of collocation methods. JNAIAM. J. Numer. Anal. Ind. Appl. Math. 5(1–2), 73–84 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Hairer, E.: Challenges in geometric numerical integration. In: Trends in Contemporary Mathematics, pp. 125–135. Springer (2014)Google Scholar
  29. 29.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38(2), 414–441 (electronic) (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hairer, E., Lubich, C.: Oscillations over long times in numerical Hamiltonian systems. In: Highly oscillatory problems, Vol. 366 of London Mathematical Society Lecture Note Series, pp. 1–24. Cambridge University Press, Cambridge (2009)Google Scholar
  31. 31.
    Hairer, E., Lubich, C.: Long-term analysis of the Störmer-Verlet method for Hamiltonian systems with a solution-dependent frequency. Numerische Mathematik 134(1), 119–138 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer-Verlet method. Acta Numer. 12, 399–450 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Vol. 31 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)Google Scholar
  34. 34.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, Vol. 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993)Google Scholar
  35. 35.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44(3), 1026–1048 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Iavernaro, F., Trigiante, D.: High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4(1–2), 87–101 (2009)Google Scholar
  38. 38.
    Iserles, A.: Solving linear ordinary differential equations by exponentials of iterated commutators. Numer. Math. 45(2), 183–199 (1984)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Iserles, A.: On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42(3), 561–599 (2002)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Iserles, A., Nørsett, S.P.: On the solution of linear differential equations in Lie groups. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 983–1019 (1999)Google Scholar
  41. 41.
    Iserles, A., Nørsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2057), 1383–1399 (2005)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numer. 9, 215–365 (2000)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Iserles, A., Quispel, G.R.W., Tse, P.S.P.: B-series methods cannot be volume-preserving. BIT 47(2), 351–378 (2007)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Kahan, W.: Unconventional methods for trajectory calculations. Department of Mathematics, University of California at Berkeley (1993)Google Scholar
  45. 45.
    Kahan, W., Li, R.-C.: Unconventional schemes for a class of ordinary differential equations-with applications to the Korteweg-de Vries equation. J. Comput. Phys. 134(2), 316–331 (1997)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Lasagni, F.M.: Canonical Runge-Kutta methods. Z. Angew. Math. Phys. 39(6), 952–953 (1988)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Laskar, J.: Chaos in the solar system. Ann. Henri Poincaré 4(suppl. 2), S693–S705 (2003)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Laskar, J., Fienga, A., Gastineau, M., Manche, H.: La2010: a new orbital solution for the long-term motion of the Earth. Astron. Astrophys. 532, A89 (2011)CrossRefGoogle Scholar
  49. 49.
    Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)Google Scholar
  50. 50.
    Levin, D.: Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 38(158), 531–538 (1982)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lord, G., Malham, S.J.A., Wiese, A.: Efficient strong integrators for linear stochastic systems. SIAM J. Numer. Anal. 46(6), 2892–2919 (2008)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Mackey, D.S., Mackey, N., Tisseur, F.: Structured factorizations in scalar product spaces. SIAM J. Matrix Anal. Appl. 27(3), 821–850 (2005)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)MathSciNetCrossRefGoogle Scholar
  55. 55.
    McLachlan, R.: Comment on: Poisson schemes for Hamiltonian systems on Poisson manifolds [Comput. Math. Appl. 27 (1994), no. 12, 7–16; MR1284126 (95d:65069)] by W.J. Zhu and M.Z. Qin, Comput. Math. Appl. 29(3), 1 (1995)Google Scholar
  56. 56.
    McLachlan, R.I.: Composition methods in the presence of small parameters. BIT 35(2), 258–268 (1995)MathSciNetCrossRefGoogle Scholar
  57. 57.
    McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002)MathSciNetCrossRefGoogle Scholar
  58. 58.
    McLachlan, R.I., Modin, K., Munthe-Kaas, H., Verdier, O.: B-series are exactly the affine equivariant methods. Numerische Mathematik 133(3), 599–622 (2016)MathSciNetCrossRefGoogle Scholar
  59. 59.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021–1045 (1999)MathSciNetCrossRefGoogle Scholar
  60. 60.
    McLachlan, R.I., Quispel, G.R.W., Turner, G.S.: Numerical integrators that preserve symmetries and reversing symmetries. SIAM J. Numer. Anal. 35(2), 586–599 (1998)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (electronic) (2003)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Morbidelli, A.: Modern Celestial Mechanics. Gordon & Breach, London (2002a)Google Scholar
  63. 63.
    Morbidelli, A.: Modern integrations of solar system dynamics. Annu. Rev. Earth Planet. Sci. 30, 89–112 (2002b)CrossRefGoogle Scholar
  64. 64.
    Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Munthe-Kaas, H.: Runge-Kutta methods on Lie groups. BIT 38(1), 92–111 (1998)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Munthe-Kaas, H., Verdier, O.: Aromatic Butcher series. Found. Comput. Math. 16, 183–215 (2016)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1(3), 297–324 (2001)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Murua, A., Sanz-Serna, J.M.: Word series for dynamical systems and their numerical integrators, Technical report, Universidad Carlos III de Madrid (2015). arXiv:1502.05528v2 [math.NA]MathSciNetCrossRefGoogle Scholar
  69. 69.
    Murua, A., Sanz-Serna, J.M.: Word series for dynamical systems and their numerical integrators. Foundations of Computational Mathematics 17(3), 675–712 (2017)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Neĭshtadt, A.I.: The separation of motions in systems with rapidly rotating phase. Prikl. Mat. Mekh. 48(2), 197–204 (1984)MathSciNetGoogle Scholar
  71. 71.
    Olver, S.: On the quadrature of multivariate highly oscillatory integrals over non-polytope domains. Numer. Math. 103(4), 643–665 (2006)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Owren, B., Marthinsen, A.: Integration methods based on canonical coordinates of the second kind. Numer. Math. 87(4), 763–790 (2001)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Petrera, M., Pfadler, A., Suris, Y.B.: On integrability of Hirota-Kimura type discretizations. Regul. Chaotic Dyn. 16(3–4), 245–289 (2011)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Quispel, G.R.W.: Volume-preserving integrators. Phys. Lett. A 206(1–2), 26–30 (1995)Google Scholar
  75. 75.
    Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(4), 045206, 7 (2008)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Ramos, A.G.C.P., Iserles, A.: Numerical solution of Sturm-Liouville problems via Fer streamers. Numer. Math. 131(3), 541–565 (2015)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Sanz-Serna, J.M.: Runge-Kutta schemes for Hamiltonian systems. BIT 28(4), 877–883 (1988)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Sanz-Serna, J.M.: An unconventional symplectic integrator of W. Kahan. Appl. Numer. Math. 16(1–2), 245–250. A Festschrift to honor Professor Robert Vichnevetsky on his 65th birthday (1994)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems, Vol. 7 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1994)Google Scholar
  80. 80.
    Shang, Z.J.: Generating functions for volume-preserving mappings and Hamilton-Jacobi equations for source-free dynamical systems. Sci. China Ser. A 37(10), 1172–1188 (1994)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Singh, P.: Algebraic theory for higher-order methods in computational quantum mechanics, Technical report, DAMTP, University of Cambridge (2015). arXiv:1510.06896v1 [math.NA]
  82. 82.
    Skokos, S.K., Gottwald, G., Laskar, J.: Chaos, Detection and Predictability, Springer, p. 18 (2016)Google Scholar
  83. 83.
    Suris, Y.B.: Preservation of symplectic structure in the numerical solution of Hamiltonian systems. In: Numerical Solution of Ordinary Differential Equations (Russian), Akad. Nauk SSSR, Inst. Prikl. Mat., Moscow, pp. 148–160, 232, 238–239 (1988)Google Scholar
  84. 84.
    Sussman, G.J., Wisdom, J.: Chaotic evolution of the solar system. Science 257(5066), 56–62 (1992)MathSciNetCrossRefGoogle Scholar
  85. 85.
    Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146(6), 319–323 (1990)MathSciNetCrossRefGoogle Scholar
  86. 86.
    van der Kamp, P.H., Kouloukas, T.E., Quispel, G.R.W., Tran, D.T., Vanhaecke, P.: Integrable and superintegrable systems associated with multi-sums of products. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470(2172), 20140481, 23 (2014)Google Scholar
  87. 87.
    Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Heidelberg; Science Press Beijing, Beijing (2013)CrossRefGoogle Scholar
  88. 88.
    Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.La Trobe UniversityMelbourneAustralia

Personalised recommendations