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W. Auzinger, O. Koch, M. Quell:
"Adaptive High-Order Splitting Methods for Systems of Nonlinear Evolution Equations with Periodic Boundary Conditions";
Numerical Algorithms, 75 (2017), 1; 261 - 283.

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.

Nonlinear evolution equations · Splitting methods · Adaptive time integration · Local error · Convergence

http://dx.doi.org/10.1007/s11075-016-0206-8

http://link.springer.com/article/10.1007/s11075-016-0206-8

Project Head Othmar Koch:
Adaptives Splitting für nichtlineare Schrödingergleichungen