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

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