BIT Numerical Mathematics

, Volume 49, Issue 3, pp 487–508

Splitting methods with complex times for parabolic equations

  • F. Castella
  • P. Chartier
  • S. Descombes
  • G. Vilmart

DOI: 10.1007/s10543-009-0235-y

Cite this article as:
Castella, F., Chartier, P., Descombes, S. et al. Bit Numer Math (2009) 49: 487. doi:10.1007/s10543-009-0235-y


Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations.

We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.


Splitting Complex time steps Composition method Higher order Parabolic equations 

Mathematics Subject Classification (2000)

65M12 47D06 

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  • F. Castella
    • 1
  • P. Chartier
    • 2
  • S. Descombes
    • 3
  • G. Vilmart
    • 4
  1. 1.IRMAR and INRIA RennesUniversité de Rennes 1Rennes CedexFrance
  2. 2.INRIA Rennes and Ecole Normale Supérieure de Cachan, Antenne de BretagneBruzFrance
  3. 3.Université de Nice–Sophia AntipolisNice Cedex 02France
  4. 4.Section de mathématiquesUniversity of GenevaGenève 4Switzerland

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