Explicit complex time integrators for stiff problems

TL;DR

This paper explores the use of complex-valued time steps in numerical integration, revealing expanded stability regions and computational advantages for complex and certain real-valued stiff systems, especially the Schrödinger equation.

Contribution

It introduces the concept of complex time integrators, demonstrating their optimality for the Schrödinger equation and their benefits when combined with Projective Integration for stiff systems.

Findings

Complex time steps expand stability regions.

Complex integrators are optimal for the Schrödinger equation.

Coupling with Projective Integration benefits real-valued stiff systems.

Abstract

Most numerical methods for time integration use real-valued time steps. Complex time steps, however, can provide an additional degree of freedom, as we can select the magnitude of the time step in both the real and imaginary directions. We show that specific paths in the complex time plane lead to expanded stability regions, providing clear computational advantages for complex-valued systems. In particular, we highlight the Schr\"odinger equation, for which complex time integrators can be uniquely optimal. Furthermore, we demonstrate that these benefits extend to certain classes of real-valued stiff systems by coupling complex time steps with the Projective Integration method.