Symmetric-conjugate splitting methods for linear unitary problems

TL;DR

This paper investigates symmetric-conjugate splitting methods for linear unitary problems, demonstrating their ability to preserve unitary properties and introducing new high-order schemes tested on the Schrödinger equation.

Contribution

The paper introduces new symmetric-conjugate splitting methods up to order six that are conjugated to unitary transformations for linear differential equations.

Findings

Methods are conjugated to unitary transformations for small time steps

New high-order schemes up to order six are constructed

Methods are tested successfully on the linear Schrödinger equation

Abstract

We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schr\"odinger equation.