Walking into the complex plane to "order" better time integrators

TL;DR

This paper explores the use of complex-valued time steps in numerical integrators, enabling higher accuracy, expanded stability, and breaking traditional order barriers, especially benefiting complex systems like the Schrödinger equation.

Contribution

It introduces a method to derive complex time step paths for explicit and implicit integrators, demonstrating advantages in accuracy and stability, and breaking the Runge-Kutta order barrier.

Findings

Complex time steps improve accuracy and stability.

Paths in the complex plane can be optimized for specific methods.

Achieved 5th order accuracy with only five function evaluations.

Abstract

Most numerical methods for time integration use real time steps. Complex time steps provide an additional degree of freedom, as we can select the magnitude of the step in both the real and imaginary directions. By time stepping along specific paths in the complex plane, integrators can gain higher orders of accuracy or achieve expanded stability regions. We show how to derive these paths for explicit and implicit methods, discuss computational costs and storage benefits, and demonstrate clear advantages for complex-valued systems like the Schrodinger equation. We also explore how complex time stepping also allows us to break the Runge-Kutta order barrier, enabling 5th order accuracy using only five function evaluations for real-valued differential equations.