Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions
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.
KeywordsNonlinear evolution equations Splitting methods Adaptive time integration Local error Convergence
Mathematics Subject Classification (2010)65J10 65L05 65M12 65M15
- 14.Auzinger, W., Hofstätter, H., Ketcheson, D., Koch, O.: Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: construction of optimized schemes and pairs of schemes, BIT Numer. Math., published online 28 July 2016Google Scholar
- 15.Auzinger, W., Koch, O.: Coefficients of various splitting methods, http://www.asc.tuwien.ac.at/~winfried/splitting/
- 18.Gray, P., Scott, S.: Chemical waves and instabilities. Clarendon, Oxford (1990)Google Scholar
- 21.Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill (1987)Google Scholar
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.