Splitting methods with complex coefficients for linear and nonlinear evolution equations

Abstract

This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The standard class of splitting methods involving real coefficients is contrasted with an alternative approach that relies on the incorporation of complex coefficients. In view of long-term computations for linear evolution equations, it is expedient to distinguish symmetric, symmetric-conjugate, and alternating-conjugate schemes. The scope of applications comprises high-order reaction-diffusion equations and complex Ginzburg-Landau equations, which are of relevance in the theories of patterns and superconductivity. Time-dependent Gross-Pitaevskii equations and their parabolic counterparts, which model the dynamics of Bose-Einstein condensates and arise in ground state computations, are formally included as special cases. Numerical experiments confirm the validity of theoretical stability conditions and global error bounds as well as the benefits of higher-order complex splitting methods in comparison with standard schemes.

Figures (7)

• Figure 1: Fourth- and sixth-order exponential operator splitting methods applied in numerical experiments, see Methods 3, 4, 8, 10, 11, 12 in Table \ref{['tab:Table1']}.
• Figure 2: Short-term integrations of a one-dimensional reaction-diffusion equation (left) and a three-dimensional high-order reaction-diffusion equation (right) by real and complex fourth- and sixth-order exponential operator splitting methods, see Figure \ref{['fig:Figure1']}. Global errors versus time stepsizes (first row) and total numbers of spectral transforms (second row), respectively. The complex exponential operator splitting methods remain stable and retain their classical orders, whereas the real splitting method by Yoshida suffers from severe instabilities.
• Figure 3: Short-term integration of complex Ginzburg--Landau equations by real and complex fourth- and sixth-order exponential operator splitting methods, see Figure \ref{['fig:Figure1']}. Global errors versus time stepsizes. Left: Standard implementation for a one-dimensional problem. The complex exponential operator splitting methods remain stable, whereas the real splitting method by Yoshida becomes unstable for larger time stepsizes. The black reference line corresponds to slope four and shows that all real and complex schemes suffer from severe order reductions. Right: Correct implementation for a related three-dimensional problem. All complex exponential operator splitting methods remain stable and retain their nonstiff orders.
• Figure 4: Corresponding results for related equations of parabolic and Schrödinger type. For a standard implementation of higher-order complex exponential operator splitting methods, severe order reductions are observed.
• Figure 5: Short-term integrations of three-dimensional complex Ginzburg--Landau equations by real and complex fourth- and sixth-order exponential operator splitting methods, see Figure \ref{['fig:Figure1']}. Global errors versus time stepsizes (first row) and total numbers of spectral transforms (second row), respectively. For $\alpha_1 = 1 + \mathrm{i}$, all complex exponential operator splitting methods remain stable and retain their classical orders. For $\alpha_1 = 1 + 10 \, \mathrm{i}$, the stability conditions \ref{['eq:StabilityConditions']} have an impact on the schemes with complex coefficients $(a_j)_{j=1}^s$, whereas the schemes involving non-negative coefficients $(a_j)_{j=1}^s$ maintain stability.

Theorems & Definitions (1)

• Theorem 1