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

TL;DR

This work addresses efficient time integration for the stiff, multidimensional complex Ginzburg–Landau equation on Cartesian domains by deploying high-order exponential-type integrators (splitting and Lawson) with constant time steps. It leverages a tensor-based $\mu$-mode approach or Fourier pseudospectral discretization to compute actions of the linear operator and its exponential without forming the full matrix, enabling scalable 2D and 3D simulations of cubic, cubic–quintic, and coupled nonlinearities. Across extensive experiments, the exponential-type methods often outperform traditional explicit Runge–Kutta schemes and Strang splitting, particularly for stringent accuracy, with splitting schemes frequently delivering the best efficiency in many settings. The results have practical implications for simulating dissipative solitons, necklace-ring patterns, and related phenomena, and point to potential extensions to variable-step, fractional, and more complex models.

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 $μ$-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.

Figures (11)

• Figure 1: Results for the simulation of the 2D cubic CGL equation \ref{['eq:ccgl']} with homogeneous Dirichlet boundary conditions and $N=256^2$ spatial discretization points. The number of time steps for each integrator is reported in Table \ref{['tab:cgl_cubic_2D_FD']}. The final simulation time is $T=6$.
• Figure 2: Dynamics of $\lvert u \rvert$ for the 2D cubic CGL equation \ref{['eq:ccgl']} with homogeneous Dirichlet boundary conditions and $N=256^2$ spatial discretization points. The time integrator is split4. The final times are $T=30$ (top left), $T=60$ (top right), $T=120$ (bottom left), and $T=240$ (bottom right) with a number of time steps equal to $m=50$, $m=100$, $m=200$, and $m=400$, respectively. The wall-clock times of the simulations are $0.54$, $1.04$, $1.76$, and $3.62$ seconds, respectively.
• Figure 3: Results for the simulation of the 2D cubic CGL equation \ref{['eq:ccgl']} with periodic boundary conditions and $N=256^2$ spatial discretization points. The number of time steps for each integrator is reported in Table \ref{['tab:cgl_cubic_2D_fourier']}. The final simulation time is $T=6$.
• Figure 4: Dynamics of $\lvert u \rvert$ for the 2D cubic CGL equation \ref{['eq:ccgl']} with periodic boundary conditions and $N=256^2$ spatial discretization points. The time integrator is split4. The final times are $T=30$ (top left), $T=60$ (top right), $T=120$ (bottom left), and $T=240$ (bottom right) with a number of time steps equal to $m=50$, $m=100$, $m=200$, and $m=400$, respectively. The wall-clock times of the simulations are $0.33$, $0.59$, $1.15$, and $2.20$ seconds, respectively.
• Figure 5: Results for the simulation of the 3D cubic CGL equation \ref{['eq:ccgl3D']} with homogeneous Dirichlet--Neumann boundary conditions and $N=128^3$ spatial discretization points. The number of time steps for each integrator is reported in Table \ref{['tab:cgl_cubic_3D_FD']}. The final simulation time is $T=10$.