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

Winfried Auzinger, Othmar Koch, Benson K. Muite, Michael Quell

Published 2016-04-05Version 1

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. Numerical examples computed for the Gray-Scott, Gierer-Meinhardt and Van der Pol equations 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. Finally we demonstrate that our methods can be extended also to nonlinear wave equations by solving the Klein-Gordon and Sine-Gordon equations.