Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators

TL;DR

This paper introduces a Julia package for symbolic manipulation of nonlinear evolution equations, aiding in the analysis of operator splitting methods and proving their convergence and error estimator correctness.

Contribution

The paper presents a novel Julia package that automates symbolic computations for nonlinear evolution equations, facilitating rigorous analysis of splitting methods.

Findings

Proved convergence of the basic splitting method

Established asymptotic correctness of a defect-based error estimator

Demonstrated package effectiveness on multiple examples

Abstract

We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples.