Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

TL;DR

This paper introduces new higher-order splitting methods for efficiently solving nonlinear evolution equations, focusing on constructing optimized schemes and pairs suitable for adaptive integration.

Contribution

It develops a systematic approach for generating polynomial systems to construct and optimize splitting schemes, including pairs for adaptive methods.

Findings

New higher-order splitting schemes constructed via polynomial systems

Pairs of schemes designed for adaptive integration

Enhanced efficiency in solving nonlinear evolution equations

Abstract

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.