Splitting methods with complex coefficients for linear and nonlinear evolution equations

TL;DR

This paper explores exponential operator splitting methods with complex coefficients for solving evolution equations, demonstrating their stability and accuracy benefits over traditional real-coefficient schemes through theoretical analysis and numerical experiments.

Contribution

It introduces novel complex coefficient splitting methods and analyzes their stability and error bounds for both linear and nonlinear evolution equations.

Findings

Higher-order complex splitting methods outperform standard schemes.

Numerical experiments confirm stability and accuracy improvements.

Applications include reaction-diffusion, Ginzburg-Landau, and Bose-Einstein condensate models.

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.