Splitting methods

Acta Numerica

Volumn 11, Issue , 2002, Pages 341-434

  McLachlan, Robert I ^a   Quispel, G Reinout W ^b  

^a MASSEY UNIVERSITY   (New Zealand)

^b LA TROBE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85095839561     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492902000053     Document Type: Article

References (142)

1. G. P. Agrawal (1989), Nonlinear Fiber Optics, Academic Press, Boston.
2. M. P. Allen and D. J. Tildesley (1987), Computer Simulation of Liquids, Oxford University Press, Oxford.
3. K. A. Bagrinovskii and S. K. Godunov (1957), Difference schemes for multidimensional problems, Dokl. Acad. Nauk SSSR 115, 431-433.
4. D. A. Barry, K. Bajracharya and C. T. Miller (1996), Alternative split-operator approach for solving chemical reaction groundwater transport models, Adv. Water Res. 19, 261-275.
5. E. Barth, B. Leimkuhler and S. Reich (1999), A time-reversible variable-stepsize integrator for constrained dynamics, SIAM J. Sci. Comput. 21, 1027-1044.
6. S. Blanes (2001), High order numerical integrators for differential equations using composition and processing of low order methods, Appl. Numer. Math. 37, 289-306.
7. S. Blanes and P. C. Moan (2000), Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. Preprint.
8. S. Blanes and P. C. Moan (2001), Splitting methods for non-autonomous Hamilto-nian equations, J. Comput. Phys. 170, 205-230.
9. S. Blanes, F. Casas and J. Ros (2000a), Symplectic integrators with processing: A general study, SIAM J. Sci. Comput. 21, 711-727.
10. S. Blanes, F. Casas and J. Ros (2000b), Processing symplectic methods for near-integrable Hamiltonian systems, Celest. Mech. Dyn. Astr. 77, 17-35.
11. S. Blanes, F. Casas and J. Ros (2001), High-order Runge-Kutta-Nyström geometric methods with processing, Appl. Numer. Math. 39, 245-259.
12. P. Bochev and C. Scovel (1994), On quadratic invariants and symplectic structure, BIT 34, 337-345.
13. H. W. Broer, G. B. Huitema and M. B. Sevryuk (1996), Quasi-Periodic Motions in Families of Dynamical Systems, Vol. 1645 of Lecture Notes in Mathematics, Springer, Berlin.
14. C. J. Budd and A. Iserles (1999), Geometric integration: Numerical solution of differential equations on manifolds, R. Soc. London Philos. Trans. A 357, 943-1133.
15. C. J. Budd and M. D. Piggott (2002), Geometric integration and its applications, lecture notes available at http://www.bath.ac.uk/~mascjb/home.html.
16. J. Butcher (1969), The effective order of Runge-Kutta methods, in Conf. Numer. Sol. Diff. Eqs., Vol. 109 of Lecture Notes in Mathematics, Springer, Berlin, pp. 133-139.
17. M. P. Calvo, A. Iserles and A. Zanna (1997), Numerical solution of isospectral flows, Math. Comput. 66, 1461-1486.
18. E. Cartan (1909), Les groupes de transformation continus, infinis, simples, Ann. Sci. Ecole Norm. Sup. 26, 93-161; Ouvres Complètes d'Elie Cartan, Springer, Berlin, 1984.
19. P. J. Channell and F. R. Neri (1996), A brief introduction to symplectic integrators, in Integration Algorithms and Classical Mechanics (J. E. Marsden, G. W. Patrick and W. F. Shadwick, eds), AMS, pp. 45-58.
20. P. J. Channell and J. C. Scovel (1990), Symplectic integration of Hamiltonian systems, Nonlinearity 3, 231-259.
21. S. A. Chin and D. W. Kidwell (2000), Higher-order force gradient symplectic algorithms, Phys. Rev. E 62, 8746-8752, Part B.
22. B. V. Chirikov (1979), A universal instability of many-dimensional oscillators systems, Phys. Rep. 52, 263-379.
23. M. Crandall and A. Majda (1986), The method of fractional steps for conservation laws, Numer. Math. 34, 285-314.
24. J. B. De Lambre (1790-91), De l'usage du calcul différentiel dans la construction des tables astronomiques, Mémoires de L'Académie Royale des Sciences de Turin 5, 143-180.
25. H. De Raedt (1987), Product formula algorithms for solving the time dependent Schrödinger equation, Comput. Phys. Rep. 7, 1-72.
26. H. De Raedt and B. De Raedt (1983), Applications of the generalized Trotter formula, Phys. Rev. A 28, 3575-3580.
27. H. De Raedt and A. Lagendijk (1985), Monte Carlo simulations of quantum statistical lattice models, Phys. Rep. 127, 233-307.
28. R. Descartes (1637), Discours de la Méthode, Leiden. English edition Discourse on Method, translated by L. J. Lafleur: Bobbs-Merrill (1964), p. 15.
29. R. Devogelaere (1956), Methods of integration which preserve the contact transformation property of the Hamiltonian equations, Report 4, Center of Numerical Analysis, University of Notre Dame.
30. L. Dieci, R. D. Russell and E. S. van Vleck (1997), On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal. 34, 402-423.
31. V. Dorodnitsyn (1996), Continuous symmetries of finite-difference evolution equations and grids, in Symmetries and Integrability of Difference Equations (Estérel, PQ, 1994), AMS, Providence, pp. 103-112.
32. A. J. Dragt (2002) Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics. Technical Report, Physics Department, University of Maryland.
33. A. J. Dragt and D. T. Abell (1996), Symplectic maps and computation of orbits in particle accelerators, in Integration Algorithms and Classical Mechanics (J. E. Marsden, G. W. Patrick and W. F. Shadwick, eds), AMS, Providence, pp. 59-85.
34. A. Dullweber, B. Leimkuhler and R. I. McLachlan (1997), Split-Hamiltonian methods for rigid body molecular dynamics, J. Chem. Phys. 107, 5840-5851.
35. J. Dutka (1997), A note on 'Kepler's equation', Arch. Hist. Exact Sci. 51, 59-65.
36. K. Feng (1992), Formal power series and numerical algorithms for dynamical systems, in Proc. Conf. Scientific Computation Hangzhou, 1991 (T. Chan and Z.-C. Shi, eds), World Scientific, Singapore, pp. 28-35.
37. K. Feng (1993), Symplectic, contact and volume-preserving algorithms, in Proc 1st China-Japan Conf. Numer. Math. (Z. C. Shi, T. Ushijima, eds), World Scientific, Singapore.
38. K. Feng (1998), Contact algorithms for contact dynamical systems, J. Comput. Math. 16, 1-14.
39. K. Feng and D.-L. Wang (1994), Dynamical systems and geometric construction of algorithms, in Vol. 163 of Contemporary Mathematics, AMS, Providence, pp. 1-32.
40. K. Feng and D.-L. Wang (1998), Variations on a theme by Euler, J. Comput. Math. 16, 97-106.
41. E. Forest (1998), Beam Dynamics, Harwood Academic Publishers.
42. E. Forest and R. Ruth (1990), Fourth-order symplectic integration, Physica D 43, 105-117.
43. T. Fukushima (1999), Fast procedure solving universal Kepler's equation, Cel. Mech. Dyn. Astron. 75, 201-226.
44. Z. Ge and J. Marsden (1988), Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133, 135-139.
45. B. Georgeot and D. L. Shepelyansky (2001), Stable quantum computation of unstable classical chaos, Phys. Rev. Lett. 86, 5393-5396.
46. R. Glowinski, T. W. Pan, T. I. Hesla and D. D. Joseph (1999), A distributed Lagrange multiplier fictitious domain method for particulate flows, Int. J. Multiphase Flow 25, 755-794.
47. S. K. Godunov (1999), Reminiscences about difference schemes, J. Comput. Phys. 163, 6-25.
48. J. Guckenheimer and P. Holmes (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Applied Mathematical Sciences, Springer, New York.
49. E. Hairer, Ch. Lubich and G. Wanner (2002), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, Springer.
50. R. H. Hardin and F. D. Tappert (1973), Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Review 15, 423.
51. D. J. Hardy, D. I. Okunbor and R. D. Skeel (1999), Symplectic variable step size integration for N-body problems, Appl. Numer. Math. 29, 19-30.
52. J. Hilgert, K. Hofmann, K. Heinrich and J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford University Press, New York.
53. A. Iserles (1984), Composite methods for numerical solution of stiff systems of ODE's, SIAM J. Numer. Anal. 21, 340-351.
54. A. Iserles and G. R. W. Quispel (2002), Stable compositions for parabolic PDEs and stiff ODEs. In preparation.
55. A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna (2000), Lie group methods, Acta Numerica, Vol. 9, Cambridge University Press, pp. 215-365.
56. S.-J. Kang and M.-H. Kim (1996), Free Lie algebras, generalized Witt formula, and the denominator identity, J. Algebra 183, 560-594.
57. K. H. Karlsen and K.-A. Lie (1999), An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. Numer. Anal. 19, 609-635.
58. F. Klein (1893), A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, 215-249.
59. Translated by M. W. Haskell. S. Kobayashi (1972), Transformation Groups in Differential Geometry, Springer, Berlin.
60. D. A. Kofke (2000), Molecular Simulation, available from: http://www.cheme.buffalo.edu/courses/ce530/Lectures/
61. P.-V. Koseleff (1995), About approximations of exponentials, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Paris, 1995), Vol. 948 of Lecture Notes in Computer Science, Springer, Berlin, pp. 323-333.
62. A. Kværnø and B. Leimkuhler (2000), A time-reversible, regularized, switching integrator for the N-body problem, SIAM J. Sci. Comput. 22, 1016-1035.
63. J. L. Lagrange (1788), Mécanique Analytique, Chez la Veuve Disaint, Paris.
64. J. S. W. Lamb, ed. (1998), Time-reversal symmetry in dynamical systems, Physica D 112, 1-328.
65. M. Lecar, F. A. Franklin, M. J. Holman and M. W. Murray (2001), Chaos in the solar system, Ann. Rev. Astron. Astrophys. 39, 581-631.
66. B. Leimkuhler (2002), Molecular Dynamics, available from: http://www.math.ukans.edu/~leimkuhl/molecular.html
67. B. Leimkuhler and S. Reich (2001), A reversible averaging integrator for multiple time-scale dynamics, J. Comput. Phys. 171, 95-114.
68. B. J. Leimkuhler and R. D. Skeel (1994), Symplectic numerical integrators in constrained Hamiltonian systems, J. Comput. Phys. 112, 117-125.
69. B. J. Leimkuhler, S. Reich and R. D. Skeel (1996), Integration methods for molecular dynamics, in Mathematical Approaches to Biomolecular Structure and Dynamics (Minneapolis, MN, 1994), Vol. 82 of IMA Volumes in Mathematics and its Applications, Springer, New York, pp. 161-185.
70. B. P. Leonard, A. P. Lock and M. K. MacVean (1996), Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes, Monthly Weather Rev. 124, 2588-2606.
71. R. J. LeVeque (1998), Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys. 146, 346-365.
72. D. Levesque and L. Verlet (1993), Molecular dynamics and time reversibility, J. Statist. Phys. 72, 519-537.
73. O. Loos (1969), Symmetric Spaces: I General Theory; II Compact Spaces and Classification, W. A. Benjamin, Inc., New York/Amsterdam.
74. M. A. López-Marcos, J. M. Sanz-Serna and R. D. Skeel (1996), Cheap enhancement of symplectic integrators, in Numerical Analysis 1995 (Dundee, 1995), Vol. 344 of Pitman Research Notes in Mathematics, Longman, Harlow, pp. 107-122.
75. M. A. López-Marcos, J. M. Sanz-Serna and R. D. Skeel (1997), Explicit symplectic integrators using Hessian-vector products, SIAM J. Sci. Comput. 18, 223-238.
76. B. A. Malomed and M. I. Tribelsky (1984), Bifurcations in distributed kinetic systems with aperiodic instability, Physica D 14, 67-87.
77. J. E. Marsden and T. S. Ratiu (1999), Introduction to Mechanics and Symmetry, 2nd edn, Springer, New York.
78. J. E. Marsden and M. West (2001), Discrete mechanics and variational integrators, Acta Numerica, Vol. 10, Cambridge University Press, pp. 357-514.
79. R. I. McLachlan (1993), Explicit Lie-Poisson integration and the Euler equations, Phys. Rev. Lett. 71, 3043-3046.
80. R. I. McLachlan (1994), Symplectic integration of Hamiltonian wave equations, Numer. Math. 66, 465-492.
81. R. I. McLachlan (1995), On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput. 16, 151-168.
82. R. I. McLachlan (2002), Families of high-order composition methods. To appear.
83. R. I. McLachlan and M. Perlmutter (2001), Conformal Hamiltonian systems, J. Geom. Phys. 39, 276-300.
84. R. I. McLachlan and G. R. W. Quispel (2000), Numerical integrators that contract volume, Appl. Numer. Math. 34, 253-260.
85. R. I. McLachlan and G. R. W. Quispel (2001a), Six lectures on geometric integration, in Foundations of Computational Mathematics (R. DeVore, A. Iserles and E. Süli, eds), Cambridge University Press, Cambridge, pp. 155-210.
86. R. I. McLachlan and G. R. W. Quispel (2001b), What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration, Nonlinearity 14, 1689-1706.
87. R. I. McLachlan and N. Robidoux (2000), Antisymmetry, pseudospectral methods, and conservative PDEs, in Proc. Int. Conf. EQUADIFF 99 (B. Fiedler, K. Gröger and J. Sprekels, eds), World Scientific, pp. 994-999.
88. R. I. McLachlan and C. Scovel (1996), Open problems in symplectic integration, in Integration Algorithms and Classical Mechanics (J. E. Marsden, G. W. Patrick and W. F. Shadwick, eds), AMS, pp. 151-180.
89. R. I. McLachlan, G. R. W. Quispel and G. S. Turner (1998), Numerical integrators that preserve symmetries and reversing symmetries, SIAM J. Numer. Anal. 35, 586-599.
90. R. I. McLachlan, G. R. W. Quispel and N. Robidoux (1999), Geometric integration using discrete gradients, Phil. Trans. Roy. Soc. A 357, 1021-1046.
91. R. I. McLachlan, M. Perlmutter and G. R. W. Quispel (2001), On the nonlinear stability of symplectic integrators. Preprint.
92. R. I. McLachlan, M. Perlmutter and G. R. W. Quispel (2002), Lie group foliate integrators. In preparation.
93. T. Misawa (2001), A Lie algebraic approach to numerical integration of stochastic differential equations, SIAM J. Sci. Comput. 23, 866-890.
94. P. Molino (1988), Riemannian Foliations, Birkhäuser, Boston.
95. H. Munthe-Kaas and B. Owren (1999), Computations in a free Lie algebra, R. Soc. London Philos. Trans. A 357, 957-981.
96. H. Munthe-Kaas and A. Zanna (1997), Numerical integration of differential equations on homogeneous manifolds, Foundations of Computational Mathematics (Rio de Janeiro, 1997; F. Cucker and M. Shub, eds), Springer, Berlin. pp. 305-315.
97. H. Z. Munthe-Kaas, G. R. W. Quispel and A. Zanna (2002), Applications of symmetric spaces and Lie triple systems in numerical analysis. Preprint.
98. A. Murua and J. M Sanz-Serna (1999), Order conditions for numerical integrators obtained by composing simpler integrators, R. Soc. London Philos. Trans. A 357, 1079-1100.
99. J. D. Murray (1989), Mathematical Biology, Springer, Berlin.
100. F. Neri (1988), Lie algebras and canonical integration. Preprint, Department of Physics, University of Maryland.
101. G. R. W. Quispel (1995), Volume-preserving integrators, Phys. Lett. A 206, 226-230.
102. G. R. W. Quispel and D. I. McLaren (2002), Explicit volume-preserving and sym-plectic integrators for trigonometric polynomial flows. Preprint.
103. S. Reich (1993), Numerical integration of the generalized Euler equations. Technical report 93-20, Department of Computer Science, University of British Columbia.
104. S. Reich (1999), Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36, 1549-1570.
105. J. A. G. Roberts and G. R. W. Quispel (1992), Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216, 63-177.
106. R. D. Ruth (1983), A canonical integration technique, IEEE Trans. Nucl. Sci. NS-30, 2669-2671.
107. J. P. Ryckaert, G. Ciccotti and H. J. C. Berendsen (1977), Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23, 327-341.
108. D. S. Ryu, T. W. Jones and A. Frank (1995), Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multidimensional flow, Astrophys. J. 452, 785-796.
109. J. M. Sanz-Serna (1997), Geometric integration, in The State of the Art in Numerical Analysis (York, 1996), Vol. 63 of Inst. Math. Appl. Conf. Ser. New Ser., Oxford University Press, New York, pp. 121-143.
110. Z. J. Shang (1994), Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math. 12, 265-272.
111. Z. J. Shang (2000), Resonant and diophantine step sizes in computing invariant tori of Hamiltonian systems, Nonlinearity 13, 299-308.
112. Q. Sheng (1989), Solving linear partial differential equations by exponential splitting, IMA J. Numer. Anal. 9, 199-212.
113. J. Shi and Y. T. Yan (1993), Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations, Phys. Rev. E 48, 3943-3951.
114. R. D. Skeel and K. Srinivas (2000), Nonlinear stability analysis of area-preserving integrators, SIAM J. Numer. Anal. 38, 129-148 (electronic).
115. C. Störmer (1907), Sur les trajectoires des corpuscules électrisés, Arch. Sci. Phys. Nat. 24, pp. 5-18, 113-158, 221-247.
116. G. Strang (1963), Accurate partial difference methods, I: Linear Cauchy problems, Arch. Rat. Mech. 12, 392-402.
117. G. Strang (1968), On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 507-517.
118. A. M. Stuart and A. R. Humphries (1996), Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge.
119. G. J. Sussman and J. Wisdom (1992), Chaotic evolution of the solar system, Science 257, 56-62.
120. M. Suzuki (1976), Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Comm. Math. Phys. 51, 183-190.
121. M. Suzuki (1990), Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146, 319-323.
122. J. B. Swift and P. C. Hohenberg (1989), Rayleigh-Bénard convection with time-dependent boundary conditions, Phys. Rev. A (3) 39, 4132-4136.
123. M. Takahashi and M. Imada (1984), Monte Carlo calculations of quantum systems, II: Higher order correction, J. Phys. Soc. Japan 53, 3765-3769.
124. T. Tang and Z. H. Teng (1995), Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal. 32, 110-127.
125. F. D. Tappert (1974), Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, in Nonlinear Wave Motion (A. C. Newell, ed.), Vol. 15 of Lectures in Applied Mathematics, AMS, pp. 215-216.
126. F. D. Tappert (1977), The parabolic approximation method, in Wave Propagation and Underwater Acoustics (J. B. Keller and J. S. Papadakis, eds), Vol. 70 of Lecture Notes in Physics, Springer, New York, pp. 224-287.
127. F. D. Tappert and M. G. Brown (1996), Asymptotic phase errors in parabolic approximations to the one-way Helmholtz equation, J. Acoust. Soc. Amer. 99, 1405-1413.
128. J. B. Taylor (1968), unpublished.
129. H. F. Trotter (1959), On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10, 545-551.
130. Z. Tsuboi and M. Suzuki (1995), Determining equations for higher-order decompositions of exponential operators, Int. J. Mod. Phys. B 25, 3241-3268.
131. S. Turek (1996), A comparative study of time-stepping techniques for the incompressible Navier-Stokes equations: From fully implicit non-linear schemes to semi-implicit projection methods, Int. J. Numer. Meth. Fluids 22, 987-1011.
132. R. Tyson, L. G. Stern and R. J. LeVeque (2000), Fractional step methods applied to a chemotaxis model, J. Math. Biol. 41, 455-475.
133. A. van den Essen (2000), Polynomial Automorphisms and the Jacobian Conjecture, Vol. 190 of Progress in Mathematics, Birkhäuser, Basel.
134. L. Verlet (1967), Computer 'experiments' on classical fluids, I: Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159, 98-103.
135. V. Volterra (1931), Leçons sur la Théorie Mathématique de la Lutte pour la Vie, reprinted by Editions Jacques Gabay, Sceaux (1990).
136. J. Weickert, B. M. T. Romeny and M. A. Viergever (1998), Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Proc. 7, 398-410.
137. A. Weinstein (1983), The local structure of Poisson manifolds, J. Diff. Geom. 18, 523-557.
138. J. Wisdom (1982), The origin of the Kirkwood gaps: A mapping for asteroidal motion near the 3/1 commensurability, Astron. J. 87, 577-593.
139. J. Wisdom and M. Holman (1991), Symplectic maps for the N-body problem, Astron. J. 102, 1528-1538.
140. J. Wisdom, M. Holman and J. Touma (1996), Symplectic correctors, in Integration Algorithms and Classical Mechanics (J. E. Marsden, G. W. Patrick and W. F. Shadwick, eds), AMS, Providence, pp. 217-244.
141. B. Yang and S. B. Pope (1998), An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry, Combust. Flame 112, 16-32.
142. H. Yoshida (1990), Construction of higher order symplectic integrators, Phys. Lett. A 150, 262-268.