On symmetric-conjugate composition methods in the numerical integration of differential equations

TL;DR

This paper investigates symmetric-conjugate composition methods with complex coefficients for numerical differential equation integration, analyzing their qualitative property preservation, proposing new schemes up to order 8, and demonstrating their efficiency through tests.

Contribution

It introduces new symmetric-conjugate composition schemes up to order 8 and compares their efficiency with traditional methods, highlighting their advantages at larger time steps.

Findings

Higher-order schemes are more efficient.

Symmetric-conjugate methods better preserve qualitative properties.

New schemes outperform traditional compositions at larger steps.

Abstract

We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.