Efficient simulation of complex Ginzburg--Landau equations using high-order exponential-type methods

TL;DR

This paper presents high-order exponential-type numerical methods for efficiently solving complex Ginzburg--Landau equations, demonstrating improved accuracy and stability over traditional methods in multi-dimensional simulations.

Contribution

Introduction of high-order exponential splitting and Lawson methods for complex Ginzburg--Landau equations that outperform standard techniques in accuracy and computational efficiency.

Findings

High-order exponential schemes are stable for stiff models.

Methods outperform split-step and Runge-Kutta in accuracy.

Effective tensor-based implementation for multi-dimensional problems.

Abstract

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called $\mu$-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg--Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge--Kutta integrator, for stringent accuracies.