Numerical Algorithms

, Volume 75, Issue 1, pp 261–283 | Cite as

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

Open Access
Original Paper

Abstract

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.

Keywords

Nonlinear evolution equations Splitting methods Adaptive time integration Local error Convergence 

Mathematics Subject Classification (2010)

65J10 65L05 65M12 65M15 

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Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienWienAustria
  2. 2.Fakultät für MathematikUniversität WienWienAustria

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