pythOS: A Python library for solving IVPs by operator splitting

TL;DR

pythOS is a Python library that simplifies solving differential equations using operator-splitting methods, integrating various advanced techniques and interfacing with finite element tools for practical problem-solving.

Contribution

It introduces a comprehensive Python library for operator-splitting methods, including fractional-step, additive Runge--Kutta, and multi-rate methods, with integration to finite element and exponential time-integration tools.

Findings

Demonstrates effective solution of practical differential equations problems.

Provides a flexible interface for advanced operator-splitting techniques.

Showcases the advantages of operator-splitting over monolithic approaches.

Abstract

Operator-splitting methods are widespread in the numerical solution of differential equations, especially the initial-value problems in ordinary differential equations that arise from a method-of-lines discretization of partial differential equations. Such problems can often be solved more effectively by treating the various terms individually with specialized methods rather than simultaneously in a monolithic fashion. This paper describes \pythOS, a Python software library for the systematic solution of differential equations by operator-splitting methods. The functionality of \pythOS\ focuses on fractional-step methods, including those with real and complex coefficients, but it also implements additive Runge--Kutta methods, generalized additive Runge--Kutta methods, and multi-rate, and multi-rate infinitesimal methods. Experimentation with the solution of practical problems is facilitated through an interface to the \Firedrake\ library for the finite element spatial discretization of partial differential equations and further enhanced by the convenient implementation of exponential time-integration methods and fully implicit Runge--Kutta methods available from the \Irksome\ software library. The functionality of \pythOS\ as well as some less generally appreciated aspects of operator-splitting methods are demonstrated by means of examples.