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Explicit complex time integrators for stiff problems

Jithin D. George, Julian Koellermeier, Samuel Y. Jung, Niall M. Mangan

TL;DR

The authors address the challenge of efficiently integrating stiff time-dependent problems by introducing complex time stepping, which enlarges the region of absolute stability and enables larger stable time steps. They show how complex substeps yield tunable stability polynomials, deriving optimal complex coefficients for Schrödinger-type dynamics and demonstrating substantial efficiency gains in both linear and nonlinear Schrödinger equations. The work extends Projective Integration with complex inner steps, enabling stable integration of real-valued and complex-valued stiff systems with spectral gaps, including Prothero–Robinson-type tests. Numerical experiments indicate that complex time integration can double the stable step size and reduce computational cost, suggesting a promising path for efficient quantum-system simulations and other stiff problems; open-source code is provided for reproducibility.

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 Schrödinger 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.

Explicit complex time integrators for stiff problems

TL;DR

The authors address the challenge of efficiently integrating stiff time-dependent problems by introducing complex time stepping, which enlarges the region of absolute stability and enables larger stable time steps. They show how complex substeps yield tunable stability polynomials, deriving optimal complex coefficients for Schrödinger-type dynamics and demonstrating substantial efficiency gains in both linear and nonlinear Schrödinger equations. The work extends Projective Integration with complex inner steps, enabling stable integration of real-valued and complex-valued stiff systems with spectral gaps, including Prothero–Robinson-type tests. Numerical experiments indicate that complex time integration can double the stable step size and reduce computational cost, suggesting a promising path for efficient quantum-system simulations and other stiff problems; open-source code is provided for reproducibility.

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 Schrödinger 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.
Paper Structure (12 sections, 2 theorems, 38 equations, 9 figures)

This paper contains 12 sections, 2 theorems, 38 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Expanded regions of stability with complex time-steps
  3. Asymmetrically expanded regions of stability
  4. A simple example
  5. Optimally stable integrators for the Schrodinger equation
  6. Numerical example: The nonlinear Schrodinger equation
  7. The Projective Integration method with complex time steps
  8. Projective Integration
  9. Complex PI schemes for a complex-valued stiff equation
  10. Complex PI schemes for real-valued systems
  11. Discussion
  12. Acknowledgements

Key Result

Theorem 3.1

\newlabeltheorem10 For a linear system $\dot{y} = A y$, where $y \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ is a matrix whose eigenvalues $\lambda \in \mathbb{C} \setminus \mathbb{R}$ lie entirely on either the positive imaginary axis, i.e., $\lambda = +i|\lambda|$, or the negative imag

Figures (9)

  • Figure 1: Three different complex time-stepping paths (right). The 1-step first-order cFE1, 2-step second-order cFE2 and 3-step third-order cFE3 methods share the same stability regions as the FE \ref{['eq:FE']}, RK2 \ref{['eq:rk2']} and RK3 \ref{['eq:rk3']} methods (left).
  • Figure 1: Complex coefficients in optimal stability polynomials often result in asymmetrical stability regions. The 3-step third-order (cFE3) complex integrator (red curves) is not optimal for the eigenvalues (red dots, left panel). RKOpt ketcheson2020rk yields the stability polynomial in Eq. \ref{['eq:RKOPT_stab']}, corresponding to the optimal 3-stage second-order real integrator (magenta curves). An optimized 3-stage complex integrator (brown curves) achieves the stability polynomial in Eq. \ref{['eq:complexOPT_stab']} which allows the largest possible time steps for these eigenvalues.
  • Figure 1: Stability regions corresponding to the real PFE method (blue) and complex cPFE method (red), together with the location of the complex eigenvalue $\lambda$ in \ref{['eq:classic_stiff']}.
  • Figure 2: All possible 1-step first-order, 2-step second-order, and 3-step third-order complex Forward Euler methods for linear differential equations.
  • Figure 2: Eigenvalues of the linear Schrödinger equation lie along the negative imaginary axis (red dots). The optimal two-stage, two-step, first-order stability polynomials with real coefficients $\Phi_r$(Eq. \ref{['eq:stabschro_real']}, magenta) and with complex coefficients $\Phi_{c_-}$ (Eq. \ref{['eq:stabschro_complex']}, green) yield distinct stability regions. Using complex coefficients with $\Phi_{c_-}$ expands the stability region, enabling stable time steps up to twice as large as those for the real-coefficient integrator $\Phi_r$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2