Publications of Robert McLachlan

1984

1. H. Barr, M. A. Grant, and R. I. McLachlan, Proving and development of geothermal fields -- risk, strategy and economics, Applied Mathematics Division Report 116, Department of Scientific and Industrial Research, Wellington, New Zealand, 1984

1987

2. S. L. Braunstein and R. I. McLachlan, Generalized squeezing, Phys. Rev. A 35(4) (1987), 1659-1667.

1990

3. N. D. Malmuth, C. C. Wu, H. Jafroudi, R. I. McLachlan, J. D. Cole, and R. Sahu, Asymptotic theory of transonic wind tunnel wall interference, Arnold Engineering Development Center report SC5413.FR., 1990.
4. R. I. McLachlan, Separated viscous flows via multigrid, Ph.D. thesis, Caltech (1990).

1991

5. R. I. McLachlan, A steady separated viscous corner flow, J. Fluid Mech. 231 (1991), 1-34.
6. R. I. McLachlan, The boundary layer on a finite flat plate, Phys. Fluids A 3(1) (1991), 341-438.

1992

7. R. I. McLachlan, On visualizing the four-dimensional rigid body, Program in Applied Mathematics Report 141, U. Colorado at Boulder, 1992.
8. R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity 5 (1992), 541-562.

1993

9. R. I. McLachlan, Multigrid solution of the Navier-Stokes equations at high Reynolds numbers, Program in Applied Mathematics report 167, U. Colorado at Boulder, 1993.
10. P. Atela and R. I. McLachlan, A note on the charged isosceles three-body problem, Proc. Int. Conf. Diff. Eqs. EQUADIFF 91 pp. 292-297, C. Perello, C. Simo, and J. Sola-Morales, eds., World Scientific, Singapore, 1993.
11. N. D. Malmuth, H. Jafroudi, C. C. Wu, R. I. McLachlan, and J. D. Cole, Asymptotic methods for the prediction of transonic wind-tunnel wall interference, AIAA J. 31(5) (1993), 911-918.
12. R. I. McLachlan, Integrable four-dimensional symplectic maps of standard type, Phys. Lett. A. 173(3) (1993), 211-214.
13. R. I. McLachlan, Explicit symplectic splitting methods applied to PDE's, Proc. AMS-SIAM Conf. Exploiting Symmetry in Applied Mathematics, Lects. Appl. Math. 29, AMS, 1993, pp. 325-337.
14. R. I. McLachlan, Explicit Lie-Poisson integration and the Euler equations, Phys. Rev. Lett. 71 (1993), 3043-3046.

1994

15. R. I. McLachlan, A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16(4) (1994), 31-37.
16. R. I. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer. Math. 66 (1994) 465-492
17. R. I. McLachlan, The world of symplectic space, New Scientist, 19 March 1994, 32-35.
18. R. I. McLachlan and H. Segur, A note on the motion of surfaces, Phys. Lett. A 194(3) (1994), 165-172.
19. P. Atela and R. I. McLachlan, Global behaviour of the charged isosceles three-body problem, Int. J. Bifurcations Chaos 4(4) (1994), 865-884.

1995

20. R. I. McLachlan and C. Scovel, Equivariant constrained symplectic integration, J. Nonlinear Sci. 5 (1995), 233-256.
21. R. I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comp. 16, (1995), 151-168.
22. R. I. McLachlan, Comment on "Poisson schemes for Hamiltonian systems on Poisson manifolds", Comp. Math. App. 29(3) (1995), 1.
23. R. I. McLachlan, Composition methods in the presence of small parameters, BIT 35(2) (1995), 258-268.

1996

24. R. I. McLachlan, More on symplectic correctors, in Integration Algorithms and Classical Mechanics, J.E. Marsden, G.W. Patrick, and W.F. Shadwick, eds., AMS, 1996, pp. 141-149.
25. R. I. McLachlan and C. Scovel, Open problems in symplectic integration, in Integration Algorithms and Classical Mechanics, J.E. Marsden, G.W. Patrick, and W.F. Shadwick, eds., AMS, 1996, pp. 151-180.

1997

26. R. I. McLachlan and G. R. W. Quispel, Generating functions for dynamical systems with symmetries, integrals, and differential invariants, Physica D 112 (1,2) (1997), 298-309.
27. R. I. McLachlan and S. Gray, Optimal stability polynomials for splitting methods, with application to the time-dependent Schrodinger equation, Appl. Numer. Anal. 25 (1997), 275-286.
28. A. Dullweber, B. Leimkuhler, and R. I. McLachlan, Split-Hamiltonian methods for rigid body molecular dynamics, J. Chem. Phys. 107(5) (1997), 5840-5851.
29. R. I. McLachlan, I. Szunyogh, and V. Zeitlin, Hamiltonian finite-dimensional models of baroclinic instability, Phys. Lett. A 229(5) (1997), 299-306.
    

    1998

30. R. I. McLachlan, G. R. W. Quispel, and G. S. Turner, Numerical integrators that preserve symmetries and reversing symmetries, SIAM J. Numer. Anal. 35 (1998), 586-599.
31. F. Dupont, R. I. McLachlan, and V. Zeitlin, On a possible mechanism of anomalous diffusion by Rossby waves, Phys. Fluids 10(12) (1998), 3185-3193.
32. R. I. McLachlan, G. R. W. Quispel, and N. Robidoux, Unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals, Phys. Rev. Lett. 81 (1998), 2399-2403.
    

    1999

33. F. Dupont, R. I. McLachlan, and V. Zeitlin, On a possible mechanism of anomalous diffusion in geophysical turbulence, Fundamental problematic issues in turbulence (Monte Verita, 1998), pp. 203-209, Trends Math., Birkhauser, Basel, 1999.
34. R. I. McLachlan, G. R. W. Quispel, and N. Robidoux, Geometric integration using discrete gradients, Phil. Trans. Roy. Soc. A 357 (1999), 1021-1046.
35. A. Iserles, R. I. McLachlan, and A. Zanna, Approximately preserving symmetries in the numerical integration of ordinary differential equations, Eur. J. Appl. Math. 10(5) (1999), 419-445.
36. R. I. McLachlan, Area preservation in computational fluid dynamics, Phys. Lett. A 264 (1999), 36-44.
    

    2000

37. R. I. McLachlan and G. R. W. Quispel, Numerical integrators that contract volume, Appl. Numer. Math. 34 (2000), 253-260.
38. R. I. McLachlan and N. Robidoux, Antisymmetry, pseudospectral methods, and conservative PDEs, EQUADIFF 99, B. Fiedler, K. Groger, and J. Sprekels, eds., World Scientific, 2000, pp. 994-999.
    

    2001

39. R. I. McLachlan and G. R. W. Quispel, Six Lectures on Geometric Integration, in Foundations of Computational Mathematics pages 155-210, ed. R. DeVore, A. Iserles, E. Suli, Cambridge University Press, Cambridge, 2001.
40. R. I. McLachlan and G. R. W. Quispel, What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration, Nonlinearity 14 (2001), 1689-1705.
41. R. I. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, J. Geom. Phys. 39(4) (2001), 276-300.
42. P. Kathirgamanathan, R. McKibbin and R. I. McLachlan, Source term estimation of pollution from an instantaneous point source, in MODSIM 2001, International Congress on Modelling and Simulation pp. 1013-1018, F. Ghassemi, P. Whetton, R. Little, and M. Littleboy, eds., Modelling and Simulation Society of Australia and New Zealand, Inc., 2001.
    

    2002

43. R. I. McLachlan, Families of high-order composition methods, Numerical Algorithms 31, (2002), 233-246.
44. R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica 11 (2002), 341-434.
45. P. Rynhart, R. McLachlan, R. McKibbin, and J. R. Jones, Mathematical modelling of granulation: Static and dynamic liquid bridges, Proceedings of the Fourth World Congress on Particle Technology (WCPT4), Sydney, 2002.
    

    2003

46. O. Gascuel, M. D. Hendy, A. Jean-Marie and R. McLachlan, The combinatorics of tandem duplication trees, Syst. Biol. 52 (2003), 110-118.
47. R. I. McLachlan, M. Perlmutter, and G. R. W. Quispel, Lie group foliations: Dynamical systems and integrators, Future Generation Computer Systems, 19 (2003) 1207-1219.
48. R. I. McLachlan, The emerging science of geometric integration, New Zealand Science Review, 2003. (Mac Word file).
49. R. I. McLachlan and G. R. W. Quispel, Geometric integration of conservative polynomial ODEs, Appl. Numer. Math. 45 (2003), 411-418.
50. R. I. McLachlan and B. N. Ryland, The algebraic entropy of classical mechanics, J. Math. Phys. 44 (2003), 3071-3087.
51. R. I. McLachlan, Spatial discretization of PDEs with integrals, IMA J. Numer. Anal. 23 (2003), 645-664.
52. A. Kitson, R. I. McLachlan, and N. Robidoux, Skew-adjoint finite difference methods on nonunform grids, New Zealand J. Math., 32 (2003) 139-159.
53. Book review of Geometric Numerical Integration by E Hairer, Ch Lubich, and G Wanner, SIAM Featured Review 45 (2003), 817-821.
54. P Kathirgamanathan, R McKibbin, and R I McLachlan (2003). Inverse modelling for identifying the origin and release rate of atmospheric pollution: An optimisation approach. in MODSIM 2003: International Congress on Modelling and Simulation: Proceedings (Volume 1), Modelling and Simulation Society of Australia and New Zealand, Canberra, ACT. pp. 64--69.
55. P R Rynhart, R I McLachlan, J R Jones, and R McKibbin, R. (2003). Solution of the Young-Laplace equation for three particles. Q.D. Nguyen, P.J. Ashman, K. Quast, & N. Swain (Eds.) in CHEMECA 2003: 31st Australasian Chemical Engineering Conference: Proceedings, University of Adelaide, Institution of Engineers, Adelaide, SA.

2004

56. U. Ascher and R. I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math. 48 (2004), 255-269.
57. R. I. McLachlan and G. R. W. Quispel, Explicit geometric integration of polynomial vector fields, BIT 44 (2004), 515-538.
58. R. I. McLachlan, M. Perlmutter, and G. R. W. Quispel, On the nonlinear stability of symplectic integrators, BIT 44 (2004), 99-117.
59. R. I. McLachlan and M. Perlmutter, Energy drift in reversible time integration, J. Phys. A 37(45) (2004), L593-L598.
    

    2005

60. R. I. McLachlan and A. Zanna, The discrete Moser-Veselov algorithm for the free rigid body, revisited, Foundations of Comput. Math. 5 (2005), 87-123.
61. U. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equations, J. Sci. Comput., 25 (1) (2005) 83-104.
    

    2006

62. I G Mason, R I McLachlan and D T Gerard, A double exponential model for biochemical oxygen demand, Bioresource Technology 97 (2) (2006), 273-282.
63. R I McLachlan and G R W Quispel, Geometric integrators for ODEs, J. Phys. A 39(19) 2006, 5251-5286.
64. R McLachlan and S Marsland, Kelvin--Helmholtz instability of momentum sheets in the Euler equations for planar diffeomorphisms, SIAM J. Appl. Dyn. Sys. 5 (2006), 726--758.
65. R I McLachlan and D R O'Neale, Geometric integration for a two-spin system, J. Phys. A: Math. Gen. 39 (2006) L447-L452.
66. R I McLachlan and M Perlmutter, Integrators for nonholonomic mechanical systems, J Nonlinear Sci 16(4) (2006), 283-328.
67. R. I. McLachlan and S. R. Marsland, Discrete mechanics and optimal control for image registration, ANZIAM J., 2006, vol 48, Electronic supplement 2006--2007, pp. C1-C16.
68. G R W Quispel and R I McLachlan, eds., J Phys A, Special issue on Geometric Numerical Integration, 39(19) (2006), 5251--5652. R. I. McLachlan, Integration and applications of generalized Euler equations. In E~Hairer, editor, Geometric Numerical Integration, Oberwolfach Report 14/2006, pages 68--70, 2006.

2007

69. R I McLachlan, A new implementation of symplectic Runge-Kutta methods, SIAM J Sci Comput 29(4) 2007, 1637-1649.
70. R. I. McLachlan and S. R. Marsland, N-particle dynamics of the Euler equations for planar diffeomorphisms. Dynamical Systems 22(3) 2007, 269-290.
71. Ryland B.N., McLachlan, R.I., and Frank, J., On multisymplecticity of partitioned Runge-Kutta and splitting methods, Int J Comput Math, 84(6) 2007, 847-869.
72. S Marsland and RI McLachlan, A Hamiltonian particle method for diffeomorphic image registration. In Proceedings of Information Processing in Medical Images, volume 4548 of Lecture Notes in Computer Science, pages 396-407. Springer, 2007.

2008

73. Ryland B.N., McLachlan R.I., On multisymplecticity of partitioned Runge-Kutta methods, SIAM J. Sci. Comput. 30(3) 2008, 1318-1340.
74. Hairer E, McLachlan, R I, Razakarivony A, Achieving Brouwer's law with implicit Runge-Kutta methods, BIT 48(2) 2008, 231-244.
75. McLachlan, R.I., H Z Munthe-Kaas, G R W Quispel, and A Zanna, Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields, Foundations Comput. Math. 8 2008, 335-355.
76. E Celledoni, R I McLachlan, D I McLaren, B Owren, G R W Quispel, and W Wright, Energy-preserving methods and B-series, Proc. 21^st Nordic Seminar on Computational Mathematics, 2008, 4pp.

2009

77. E Hairer, R I McLachlan, and R D Skeel, On energy behaviour of the modified Takahashi-Imada method, Mathematical Modelling and Numerical Analysis 43 (2009) 631-644.
78. E Celledoni, R I McLachlan, D I McLaren, B Owren, G R W Quispel, and W Wright, Energy-preserving Runge-Kutta methods, Mathematical Modelling and Numerical Analysis 43 (2009) 645-649.
79. R I McLachlan, G R W Quispel, and P S P Tse, Linearization-preserving self-adjoint and symplectic integrators, BIT 49(1) (2009), 177-197.
80. R I McLachlan, The structure of a set of vector fields on Poisson manifolds, J. Phys. A: Math. Theor. 42 (2009) 142001.
81. R I McLachlan and X Zhang, Well-posedness of modified Camassa-Holm equations, J. Differential Equations 246 (2009) 3241-3259.
82. A J Elvin, C R Laing, R I McLachlan, and M G Roberts, Exploiting the Hamiltonian structure of a neural field model, Physica D (2009), in press.
83. D R O'Neale and R I McLachlan, Reconsidering trigonometric integrators, ANZIAM Journal 50 (2009), 3241-3259.

    2010

84. D R J O'Neale and R I McLachlan, Preservation and destruction of periodic orbits by symplectic integrators, Numerical Algorithms 53 (2010), 343-362.
85. E Celledoni, R I McLachlan, B Owren, and G R W Quispel, On conjugate B-series and their geometric structure, Journal of Numerical Analysis, Industrial and Applied Mathematics 5 (2010), 343-362.
86. E Celledoni, R I McLachlan, B Owren, and G R W Quispel, Energy-preserving integrators and the structure of B-series, Foundations of Computational Mathematics, 10 (2010) 673-693.
87. K Modin, M Perlmutter, S Marsland, R McLachlan, Geodesics on Lie groups: Euler equations and totally geodesic subgroups, Res. Lett. Inf. Math. Sci., Massey University 14 (2010), 79-106.

    2011

88. R I McLachlan, Y Sun, and P S P Tse, Linear stability of partitioned Runge-Kutta methods, SIAM J Numer. Anal., 49 (2011), 232-263.
89. K Modin, M Perlmutter, S R Marsland, and R I McLachlan, On Euler-Arnold equations and totally geodesic subgroups, J Geom Phys 61 (2011), 1446-1461.
90. R I McLachlan and X Y Zhang, Asymptotic profiles for modified Camassa-Holm equations, SIAM J Appl. Dyn. Sys. 10 (2011), 452-468.
91. S Marsland, R McLachlan, K Modin, and M Perlmutter, On a geodesic equation for planar conformal template matching, Proc. Third Int. Workshop Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability (2011), 52-63.

    2012

92. S Marsland, R I McLachlan, K Modin, and M Perlmutter, Geodesic warps by conformal mappings, International Journal of Computer Vision, Vision (2012), 1-11, DOI 10.1007/s11263-012-0584-x.
93. E Celledoni, V Grimm, R I McLachlan, D I McLaren, D R J O'Neale, B Owren, and G R W Quispel, Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method, J Comput Phys, 231 (2012), 6770-6789.

2013

94. E Celledoni, R I McLachlan, G R W Quispel, and B Owren, Geometric properties of Kahan's method, J. Phys. A 46(2) (2013), 025201
95. R I McLachlan, K Modin, O Verdier, and M Wilkins, Symplectic integrators for index one constraints, SIAM J Sci Comput 35(5) (2013), A2150-A2162.

2014

96. R I McLachlan and G R W Quispel, Discrete gradient methods have an energy conservation law, Disc. Cont. Dyn. S. A, 34(3) (2014), 1099-1104. doi:10.3934/dcds.2014.34.1099
97. S Marsland, R McLachlan, K Modin, and M Perlmutter, On conformal variational problems and free boundary continua, J. Phys. A 47 (2014) 145204.
98. R I McLachlan, K Modin, and O Verdier, Collective symplectic integrators, Nonlinearity 27(6) (2014), 1525-1542.
99. R I McLachlan, K Modin, and O Verdier, Collective Lie-Poisson integrators on R^3, IMA J. Numer. Anal. (2014).
100. R I McLachlan and A Stern, Modified trigonometric integrators, SIAM J. Numer. Anal. 52(3) (2014), 1378-1397.
101. R I McLachlan, K Modin, O Verdier, and M Wilkins, Geometric generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339-370.
102. R I McLachlan, K Modin, and O Verdier, Symplectic integrators for spin systems, Phys. Rev. E 89(6), 061301, 2014.
103. R I McLachlan, B N Ryland, and Y Sun, High order multisymplectic Runge-Kutta methods, SIAM J. Sci. Comput. 36(5), A2199-A2226, 2014.
104. E Celledoni, R I McLachlan, D I McLaren, B Owren, and G R W Quispel, Integrability properties of Kahan's method, J Phys. A 47(36) (2014) 365202.



2015

105. R I McLachlan and M C Wilkins, The multisymplectic diamond scheme, SIAM J Sci. Comput. 37(1) (2015) A369-A390.
106. R I McLachlan, K Modin, H Munthe-Kaas, and K Modin, B-series are exactly the affine equivariant methods, Numer. Math. 133 (2016), 599-622.
107. E Celledoni, R I McLachlan, D I McLaren, B Owren, and G R W Quispel, Discretization of polynomial vector fields by polarization Proc. Roy. Soc. A 471: 20150390.

     2016

108. Geometry of discrete-time spin systems, J. Nonlinear Sci. 26(5) (2016), 1507-1523.
109. S Marsland and R I McLachlan, Möbius invariants of shapes and images, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 12 (2016), 080.
110. F McDonald, R I McLachlan, B E Moore, and G R W Quispel, Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations J. Diff. Eq. Appl. 22(7) (2016), 913-940.
111. R I McLachlan, K Modin, and O Verdier, Symmetry reduction for central force problems Eur. J. Phys. 37 (5) (2016) 055003.
112. L M Symes, R I McLachlan, and P B Blakie, Efficient and accurate methods for solving the time-dependent spin-1 Gross-Pitaevskii equation Phys. Rev. E 93(5) (2016) 053309.

     2017

113. R I McLachlan, K Modin, and O Verdier, A minimal-variable symplectic integrator on spheres Math. Comput. 86 (2017) 2325-2344.
114. V Grimm, RI McLachlan, D McLaren, GRW Quispel, CB Schönlieb, The use of discrete gradient methods for total variation type regularization problems in image processing, Journal of Physics A: Mathematical and Theoretical 50 (29) 2017, 295201
115. R I McLachlan, K Modin, H Munthe-Kaas, O Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations, Asia-Pacific Mathematics Newsletter 7 (1) 2017, 1-11.
116. Two classes of quadratic vector fields for which the Kahan discretization is integrable, E Celledoni, RI McLachlan, DI McLaren, B Owren, GRW Quispel, MI Lecture Notes of Institute of Mathematics for Industry, 74 (2017) 60-62.

     Preprints

117. R. I. McLachlan and N. Robidoux, Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations.
118. R McLachlan and X Zhang, Dynamics of the generalized Euler equations on Virasoro groups.