We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

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.