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Optimized high-order splitting methods for some classes of parabolic equations

Authors: S. Blanes, F. Casas, P. Chartier and A. Murua
Journal: Math. Comp. 82 (2013), 1559-1576
MSC (2010): Primary 65L05, 65P10, 37M15
Published electronically: December 10, 2012
MathSciNet review: 3042575
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Abstract: We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this second-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, methods of orders $ 3$ to $ 14$ by using the Suzuki-Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique and show that it is inherently bounded to order $ 14$ and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders $ 6$ and $ 8$ that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting methods with complex coefficients with positive real part by building explicitly a method of order $ 16$ as a composition of methods of order 8.

References [Enhancements On Off] (What's this?)

  • [BC05] S. Blanes and F. Casas.
    On the necessity of negative coefficients for operator splitting schemes of order higher than two.
    Appl. Num. Math., 54:23-37, 2005. MR 2134093 (2006b:65085)
  • [BCM08] S. Blanes, F. Casas, and A. Murua.
    Splitting and composition methods in the numerical integration of differential equations.
    Bol. Soc. Esp. Mat. Apl., 45:87-143, 2008. MR 2477860 (2010c:65250)
  • [BCM10] S. Blanes, F. Casas, and A. Murua.
    Splitting methods with complex coefficients.
    Bol. Soc. Esp. Mat. Apl., 50:47-61, 2010. MR 2664321 (2011e:65101)
  • [CCDV09] F. Castella, P. Chartier, S. Descombes, and G. Vilmart.
    Splitting methods with complex times for parabolic equations.
    BIT Numerical Analysis, 49:487-508, 2009. MR 2545817 (2010k:65098)
  • [CG89] M. Creutz and A. Gocksch.
    Higher-order hybrid Monte Carlo algorithms.
    Phys. Rev. Lett., 63:9-12, 1989. MR 1001905 (90e:82005)
  • [Cha03] J. E. Chambers.
    Symplectic integrators with complex time steps.
    Astron. J., 126:1119-1126, 2003.
  • [CM09] P. Chartier and A. Murua.
    An algebraic theory of order.
    M2AN Math. Model. Numer. Anal., 43:607-630, 2009. MR 2542867 (2010h:65107)
  • [DS02] S. Descombes and M. Schatzman.
    Strang's formula for holomorphic semi-groups.
    J. Math. Pures Appl., 81:93-114, 2002. MR 1994884 (2005g:35008)
  • [For89] E. Forest.
    Canonical integrators as tracking codes.
    AIP Conference Proceedings, 184:1106-1136, 1989.
  • [FT88] S. Fauve and O. Thual.
    Localized structures generated by subcritical instabilities.
    J. Phys. France, 49:1829-1833, 1988.
  • [GK96] D. Goldman and T. J. Kaper.
    $ n$th-order operator splitting schemes and nonreversible systems.
    SIAM J. Numer. Anal., 33:349-367, 1996. MR 1377257 (97a:65063)
  • [HLW06] E. Hairer, C. Lubich, and G. Wanner.
    Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition.
    Springer Series in Computational Mathematics 31. Springer, Berlin, 2006. MR 2221614 (2006m:65006)
  • [HO09a] E. Hansen and A. Ostermann.
    Exponential splitting for unbounded operators.
    Math. Comp., 78:1485-1496, 2009. MR 2501059 (2011b:65079)
  • [HO09b] E. Hansen and A. Ostermann.
    High order splitting methods for analytic semigroups exist.
    BIT Numerical Analysis, 49:527-542, 2009. MR 2545819 (2010k:65100)
  • [HKLR10] H. Holden, K.H. Karlsen, K.A. Lie, and N.H. Risebro.
    Splitting Methods for Partial Differential Equations with Rough Solutions.
    European Mathematical Society, Zürich, 2010. MR 2662342 (2011j:65002)
  • [HV03] W. Hundsdorfer and J.G. Verwer.
    Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations.
    Springer, Berlin, 2003. MR 2002152 (2004g:65001)
  • [JL00] T. Jahnke and C. Lubich.
    Error bounds for exponential operator splittings.
    BIT, 40:735-744, 2000. MR 1799313 (2001k:65143)
  • [Lub08] C. Lubich.
    On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations.
    Math. Comp., 77:2141-2153, 2008. MR 2429878 (2009d:65114)
  • [MSS99] A. Murua and J. M. Sanz-Serna.
    Order conditions for numerical integrators obtained by composing simpler integrators.
    Philos. Trans. Royal Soc. London ser. A, 357:1079-1100, 1999. MR 1694703 (2000b:65148)
  • [Ros63] H. H. Rosenbrock.
    Some general implicit processes for the numerical solution of differential equations.
    Comput. J., 5:329-330, 1962/1963. MR 0155434 (27:5368)
  • [She89] Q. Sheng.
    Solving linear partial differential equations by exponential splitting.
    IMA J. Numer. Anal., 9:199-212, 1989. MR 1000457 (90g:65163)
  • [Suz90] M. Suzuki.
    Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations.
    Phys. Lett. A, 146:319-323, 1990. MR 1059400 (91d:81005)
  • [Suz91] M. Suzuki.
    General theory of fractal path integrals with applications to many-body theories and statistical physics.
    J. Math. Phys., 32:400-407, 1991. MR 1088360 (92k:81096)
  • [TT95] T. Tang and Z-H. Teng.
    Error bounds for fractional step methods for conservation laws with source terms.
    SIAM J. Numer. Anal., 32:110-127, 1995. MR 1313707 (95m:65155)
  • [Tang98] T. Tang.
    Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms.
    SIAM J. Numer. Anal., 35:1939-1968, 1998. MR 1639974 (99f:65132)
  • [Tal08] M. Thalhammer.
    High-order exponential operator splitting methods for time-dependent Schrödinger equations.
    SIAM J. Numer. Anal., 46:2022-2038, 2008. MR 2399406 (2009b:65163)
  • [Yos90] H. Yoshida.
    Construction of higher order symplectic integrators.
    Phys. Lett. A, 150:262-268, 1990. MR 1078768 (91h:70014)

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Additional Information

S. Blanes
Affiliation: Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain

F. Casas
Affiliation: Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071 Castellón, Spain

P. Chartier
Affiliation: INRIA Rennes and Ecole Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schumann, 35170 Bruz, France

A. Murua
Affiliation: EHU/UPV, Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, 12071 Donostia/San Sebastián, Spain

Keywords: Composition methods, splitting methods, complex coefficients, parabolic evolution equations
Received by editor(s): February 1, 2011
Received by editor(s) in revised form: October 13, 2011, and December 2, 2011
Published electronically: December 10, 2012
Additional Notes: The work of the first, second and fourth authors was partially supported by Ministerio de Ciencia e Innovación (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by FEDER Funds of the European Union). Financial support from the “Acción Integrada entre España y Francia” HF2008-0105 was also acknowledged
The fourth author was additionally funded by project EHU08/43 (Universidad del País Vasco/Euskal Herriko Unibertsitatea).
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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