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

TL;DR

This paper evaluates the efficiency of adaptive high-order splitting methods for nonlinear parabolic equations with periodic boundaries, focusing on complex pattern formation and nonsmooth dynamics.

Contribution

It provides a general error analysis for splitting methods under periodic conditions and compares adaptive schemes with different operator splits using local error estimators.

Findings

Numerical examples confirm convergence of the methods.

Adaptive schemes outperform fixed-step methods in efficiency.

Splitting into two or three operators affects performance.

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.