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A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

Yong-Liang Zhao, Alexander Ostermann, Xian-Ming Gu

TL;DR

The paper tackles the computational challenge of solving the $2$D space fractional Ginzburg-Landau equation (FGLE) by combining a second-order fractional centered difference discretization with a dynamical low-rank approach. It first derives a matrix differential equation for the discretized system, then applies a stiff-linear/nonlinear Lie-Trotter split, and develops a low-rank approximation using a projector-splitting integrator to advance the nonlinear block. The authors provide rigorous convergence analysis under regularity and boundedness assumptions, and validate the method with numerical experiments that show first-order temporal and second-order spatial convergence at moderate ranks, while significantly reducing computational cost. The approach yields a robust, scalable framework for nonlinear FGLEs in 2D and offers a path to extensions to higher dimensions and other fractional PDEs.

Abstract

Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is used. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate.

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

TL;DR

The paper tackles the computational challenge of solving the D space fractional Ginzburg-Landau equation (FGLE) by combining a second-order fractional centered difference discretization with a dynamical low-rank approach. It first derives a matrix differential equation for the discretized system, then applies a stiff-linear/nonlinear Lie-Trotter split, and develops a low-rank approximation using a projector-splitting integrator to advance the nonlinear block. The authors provide rigorous convergence analysis under regularity and boundedness assumptions, and validate the method with numerical experiments that show first-order temporal and second-order spatial convergence at moderate ranks, while significantly reducing computational cost. The approach yields a robust, scalable framework for nonlinear FGLEs in 2D and offers a path to extensions to higher dimensions and other fractional PDEs.

Abstract

Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is used. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate.

Paper Structure

This paper contains 11 sections, 2 theorems, 31 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A low-rank approximation of the 2D complex FGLE
  3. The matrix differential equation
  4. The full-rank Lie-Trotter splitting method
  5. The low-rank approximation
  6. The projector-splitting integrator
  7. Convergence analysis
  8. Preliminaries for the convergence analysis
  9. Convergence analysis
  10. Numerical experiments
  11. Concluding remarks

Key Result

Theorem 3.1

Under Assumption assumption1, for $1 \leq k \leq M$, the error bound holds. Here, the constant $C_2$ depends on $C_1$, $L$ and $H$.

Figures (3)

  • Figure 1: Results for Example 1 for $(\alpha, \beta) = (1.5, 1.5)$ and $N = M = 200$. Left: Numerical rank of the LBDF2 solution as a function of $t$. Right: First $60$ singular values of the LBDF2 solution at $t = T$.
  • Figure 2: Comparison of the absolute values of the LBDF2 solution and our low-rank solution at $t = T$ for $(N, M) = (512, 200)$ and different values of $\alpha$ and $\beta$ for Example 1. Top row: The LBDF2 solution. Bottom row: The low-rank solution (rank $r = 5$).
  • Figure 3: Comparison of the absolute values of the LBDF2 solution and our low-rank solution at $t = T$ for $(N, M) = (512, 200)$ and different values of $\alpha$ and $\beta$ for Example 2. Top row: The LBDF2 solution. Bottom row: The low-rank solution (rank $r = 8$).

Theorems & Definitions (5)

  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof