Publications (papers in Scientific Journals and chapters in books)

  1. M. Antoñana, J. Makazaga, A. Murua, SIMD-vectorized implicit symplectic integrators can outperform explicit symplectic ones, Numer Algor (2026). https://doi.org/10.1007/s11075-026-02370-3
  2. M. P. Calvo, J. Makazaga, A. Murua, Taylor-Fourier Approximation, J Sci Comput 103, 69 (2025). https://doi.org/10.1007/s10915-025-02880-7
  3. S. Blanes, F. Casas, and A. Murua, Splitting methods for differential equations, Acta Numerica (2024),1-161 https://doi.org/10.1017/S0962492923000077 (Supplementary Material)
  4. M. Antoñana, E. Alberdi, J. Makazaga, A. Murua, An implicit symplectic solver for high-precision long term integrations of the Solar System, Celest Mech Dyn Astron 134, 31 (2022). https://doi.org/10.1007/s10915-025-02880-7
  5. M. Antoñana, P. Chartier, A. Murua, Majorant series for the N-body problem, arXiv:2103.12839, International Journal of Computer Mathematics, 99(1) (2022), 158-183; https://doi.org/10.1080/00207160.2021.1962848
  6. A. Murua, From Runge-Kutta Methods to Hopf Algebras of Rooted Trees, in the series Algebra and Applications 2: Combinatorial Algebra and Hopf Algebras , ISTE Ltd-Wiley 2021;
  7. M. Antoñana, P. Chartier, J. Makazaga, A. Murua, Global time-renormalization of the gravitational N-body problem, arXiv:2001.01221, SIAM J. Appl. Dyn. Syst., 19(4) (2020), 2658-2681; https://doi.org/10.1137/20M1314719,
  8. X. Tu, A. Murua, Y. Tang, New high order symplectic integrators via generating functions with its application in many-body problems, Bit Numer Math 60 (2020), 509-534; https://doi.org/10.1007/s10543-019-00785-0
  9. F. Casas, P. Chartier, A. Murua, Continuous changes of variables and the Magnus expansion, Journal of Physics Communications 3 (2019), 095014
  10. R. I. McLachlan and A. Murua, The Lie algebra of classical mechanics, arXiv:1905.07554, J. Comput. Dyn., 6 (2019), 345-360; http://dx.doi.org/10.3934/jcd.2019017
  11. A. Murua, J.M. Sanz-Serna, Hopf algebra techniques to handle dynamical systems and numerical integrators, arXiv 1702.08354, In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_22
  12. A. Murua, J.M. Sanz-Serna, Averaging and computing normal forms with word series algorithms, arXiv 1512.03601, in Discrete Mechanics, Geometric Integration and Lie-Butcher Series (DMGILBS, Madrid, May 2015), K. Ebrahimi Fard and M. Barbero Liñán eds., Springer, Berlin 2018, 115-137. DOI 978-3-030-01397-4_4.
  13. M. Antoñana, J. Makazaga, A. Murua, New Integration Methods for Perturbed ODEs Based on Symplectic Implicit Runge-Kutta Schemes with Application to Solar System Simulations, arXiv:1711.06050, Journal of Scientific Computing, ISSN 0885-7474, 2018, 76, 1 (2018), pp 630-650, DOI 10.1007/s10915-017-0634-1
  14. M. Antoñana, J. Makazaga, A. Murua, Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations, arXiv 1703.07697, Numerical Algorithms, 78, 1 (2018), pp. 63--86, DOI 10.1007/s11075-017-0367-0
  15. M. Antoñana, J. Makazaga, A. Murua, Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes, Numerical Algorithms 76, 4 (2017), pp. 861--880, doi:10.1007/s11075-017-0287-z
  16. S. Blanes, F. Casas, A. Murua, Symplectic time-average propagators for the Schödinger equation with a time-dependent Hamiltonian, The Journal of Chemical Physics 146, 114109 (2017); doi: 10.1063/1.4978410
  17. A. Murua, J.M. Sanz-Serna, Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Analysis, Theory, Methods and Applications 138 (2016), pp. 326-345.
  18. A. Murua, J.M. Sanz-Serna, Vibrational resonance: a study with high-order word-series averaging, Applied Mathematics and Nonlinear Sciences 1 (2016), pp. 239-146.
  19. A. Murua, J.M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Foundations of Computational Mathematics (2015), DOI 10.1007/s10208-015-9295-3
  20. S. Blanes, F. Casas, and A. Murua, An efficient algorithm based on splitting for the time integration of the Schrödinger equation, J. Comput. Phys., 303 (2015), pp. 396-412. (Fortran programs)
  21. J.M. Sanz-Serna and A. Murua, Formal series and numerical integrators: some history and some new techniques, in Proceedings of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), Lei Guo and Zhi-Ming eds., Higher Edication, Press, Beijing (2015), pp. 311-331.
  22. A. Murua, (2015) B-Series . In: Engquist B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_98
  23. S. Blanes, F. Casas, A. Murua (2015) Splitting Methods . In: Engquist B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_146
  24. F. Castella, P. Chartier, F. Méhats and A. Murua, Stroboscopic averaging for the nonlinear Schrödinger equation , Foundations of Computational Mathematics, Vol. 15, Issue 2 (2015), pp. 519-559.
  25. P. Chartier, A. Murua and J.M. Sanz-Serna, Higher-order averaging, formal series and numerical integration III: error bounds , Foundations of Computational Mathematics, Vol. 15, Issue 2 (2015), pp. 591-612.
  26. P. Chartier, J. Makazaga, A. Murua, and G. Vilmart, Multi-revolution composition methods for highly oscillatory differential equations, Numerische Mathematik 128, 1 (2014), pp. 167-192.
  27. A. Farrés, J. Laskar, S. Blanes, F. Casas, J. Makazaga, and A. Murua, High precision Symplectic Integrators for the Solar System. Cel. Mech. & Dyn. Astron., 116 (2013), pp. 141-174.
  28. S. Blanes, F. Casas, A. Farrés, J. Laskar, J. Makazaga, and A. Murua, New families of symplectic splitting methods for numerical integration in dynamical astronomy, Appl. Numer. Math. 68 (2013), pp. 58-72. arXiv:1208.0689v1
  29. S. Blanes, F. Casas, P. Chartier, and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comput. 82 (2013), pp. 1559-1576.
  30. F. Casas, A. Murua, and M. Nadinic, Efficient computation of the Zassenhaus formula, Computer Physics Communications 183, 11, (2012), 2386-2391.
  31. Ph. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration II: the quasi-periodic case , Foundations of Computational Mathematics, 12 (2012), 471-508.
  32. Ph. Chartier, A. Murua and J. M. Sanz-Serna, A formal series approach to averaging: exponentially small error estimates , Discrete and Continuous Dynamical Systems 32 (2012), 3009-3027.
  33. S. Blanes, F. Casas, and A. Murua, Splitting methods in the numerical integration of non-autonomous dynamical systems, RACSAM. 106 (2012), 49-66.
  34. S. Blanes, F. Casas, and A. Murua, Error analysis of splitting methods for the time dependent Schrödinger equation , SIAM J. Sci. Comput. 33 (2011), 1525-1548.
  35. M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs , Appl. Numer. Math. 61 (2011), 1077-1095.
  36. M. P. Calvo, Ph. Chartier, A. Murua, and J.M. Sanz-Serna, A stroboscopic method for highly oscillatory problems, in Numerical Analysis and Multiscale Computations, B. Engquist, O. Runborg and R. Tsai, editors, Lect. Notes Comput. Sci. Eng., Vol. 82, Springer 2011, 73-87.
  37. Ph. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series , Found. Comput. Math. 10 (2010), 695-727.
  38. S. Blanes, F. Casas, and A. Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl. 50 (2010), 47-61.
  39. P. Chartier and A. Murua, An algebraic theory of order , M2AN 43 (2009) 607-630
  40. J. Makazaga, A. Murua, A new class of symplectic integration schemes based on generating functions , Numerische Mathematik, Vol. 113, Issue 4 (2009), 631--642
  41. F. Casas, A. Murua, An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications , Journal of Mathematical Physics 50, 033513 (2009)
  42. S. Blanes, F. Casas, A. Murua, Splitting and Composition Methods in the Numerical Integration of Differential Equations , Bol. Soc. Esp. Mat. Apl. No.45 (2008), 89--145
  43. S. Blanes, F. Casas, A. Murua, On the linear stability of splitting methods , Found Comput Math 8 (2008), 357--393
  44. S. Blanes, F. Casas, A. Murua, Splitting methods for non-autonomous linear systems , Int. J. Comput. Math. 84(6) (2007), 713--727
  45. P. Chartier, A. Murua, Preserving first integrals and volume forms of additively split systems , IMA Journal of Numerical Analysis, vol. 27, 3 (2007), 381--405
  46. S. Blanes, F. Casas, A. Murua, Symplectic operator splitting methods for the time-dependent Schrödinger equation, Journal of Chemical Physics, vol. 124 (2006)
  47. P. Chartier, E. Faou, A. Murua, An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants , Numerische Mathematik, vol. 103 (2006), 575--590
  48. A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Foundations of Computational Mathematics, vol. 6 (2006), 387--426
  49. S. Blanes, F. Casas, A. Murua, Composition methods for differential equations with processing , SIAM Journal of Scientific Computing, vol. 27, No.6 (2006), 1817--1843
  50. S. Blanes, F. Casas, A. Murua, On the numerical integration of ODEs by processed methods, SIAM Journal of Numerical Analysis, vol. 42, No. 2 (2004), 531--552
  51. R.P.K. Chan , P. Chartier and A. Murua, Reversible methods of Runge-Kutta type for Index-2 Differential-Algebraic Equations, Numerische Mathematik, vol. 97, No. 3 (2004), 427--440
  52. J. Makazaga, A. Murua, New Runge-Kutta based schemes for ODEs with cheap global error estimation, Bit Numerical Mathematics, vol 43 (2003), 595-610
  53. R.P.K. Chan , P. Chartier and A. Murua, Post-projected Runge-Kutta methods for index-2 differential-algebraic equations, Applied Numerical Mathematics, 42 (2002) 77-94
  54. R. Chan, A. Murua, Extrapolation of Symplectic methods for Hamiltonian problems, Applied Numerical Mathematics 34 (2000) 189-205
  55. J. Makazaga, A. Murua, Cheap one-step global error estimation for ODEs, New Zeland Journal of Mathematics 29 (2000), 211-221
  56. A. Murua, Formal Series and Numerical integrators. Part I: Systems of ODEs and symplectic Integrators, Applied Numerical Mathematics 29 (1999), 221-251
  57. A. Murua, Formal Series and Numerical integrators. Part II: Application to index 2 differential-algebraic systems, Applied Numerical Mathematics 29 (1999), 99-113
  58. J. M. Sanz-Serna, A. Murua, Order conditions for numerical integrators obtained by composing simpler integrators, Philosophical Transactions of the Royal Society A 357 (1999), 1079-1100
  59. M. Arnold, A. Murua, Non-stiff integrators for differential-algebraic systems of index 2, Numerical Algorithms 19 (1998), 25-41
  60. A. Murua, Runge-Kutta-Nystrom methods for general second order ODEs with application to multi-body systems, Applied Numerical Mathematics (28) 2-4 (1998) 371-386
  61. A. Murua, Order conditions for partitioned symplectic methods, SIAM Journal of Numerical Analysis, Vol 34, No. 6 (1997), 2204-2211
  62. A. Murua, Partitioned half-explicit Runge-Kutta methods for differential-algebraic systems of index 2, Computing, Vol 59, No 1 (1997), 43-61
  63. A. L. Araujo, A. Murua, and J. M. Sanz-Serna, Symplectic methods based on decompositions, SIAM Journal of Numerical Analysis, Vol 34, No. 5 (1997), 1926-1947
  64. E. Hairer, A. Murua, and J. M. Sanz-Serna, The non-existence of symplectic multi-derivative Runge-Kutta methods, BIT 34 (1994), 80-87
  65. M. P. Calvo, A. Murua, and J. M. Sanz-Serna, Modified equations for ODEs, Contentemporary Mathematics, Vol 172, American Mathematical Society (1994), 63-74
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