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Strong Convergence of a Splitting Method for the Stochastic Complex Ginzburg-Landau Equation

Marvin Jans, Gabriel J. Lord, Mariya Ptashnyk

TL;DR

This paper addresses strong convergence of a splitting-based numerical scheme for the stochastic complex Ginzburg-Landau equation on the 1D torus with additive space-time noise. It employs a spectral Galerkin discretization in space and a Lie-Trotter splitting in time, proving existence, uniqueness, and moment bounds for the SCGLE and establishing strong convergence on a high-probability set with rates depending on the time step and spatial resolution. The results include convergence in probability and quantitative error bounds under a bound on the complex parameter $|\nu|\le\sqrt{3}$ and regularity assumptions, supported by rigorous energy estimates and Galerkin limits. Numerical experiments comparing ESM to exponential-splitting and taming variants confirm the predicted rates across stable and defect-turbulence regimes, validating the method's robustness for stochastic complex Ginzburg-Landau dynamics.

Abstract

We consider the numerical approximation of the stochastic complex Ginzburg-Landau equation with additive noise on the one dimensional torus. The complex nature of the equation means that many of the standard approaches developed for stochastic partial differential equations can not be directly applied. We use an energy approach to prove an existence and uniqueness result as well to obtain moment bounds on the stochastic PDE before introducing our numerical discretization. For such a well studied deterministic equation it is perhaps surprising that its numerical approximation in the stochastic setting has not been considered before. Our method is based on a spectral discretization in space and a Lie-Trotter splitting method in time. We obtain moment bounds for the numerical method before proving our main result: strong convergence on a set of arbitrarily large probability. From this we obtain a result on convergence in probability. We conclude with some numerical experiments that illustrate the effectiveness of our method.

Strong Convergence of a Splitting Method for the Stochastic Complex Ginzburg-Landau Equation

TL;DR

This paper addresses strong convergence of a splitting-based numerical scheme for the stochastic complex Ginzburg-Landau equation on the 1D torus with additive space-time noise. It employs a spectral Galerkin discretization in space and a Lie-Trotter splitting in time, proving existence, uniqueness, and moment bounds for the SCGLE and establishing strong convergence on a high-probability set with rates depending on the time step and spatial resolution. The results include convergence in probability and quantitative error bounds under a bound on the complex parameter and regularity assumptions, supported by rigorous energy estimates and Galerkin limits. Numerical experiments comparing ESM to exponential-splitting and taming variants confirm the predicted rates across stable and defect-turbulence regimes, validating the method's robustness for stochastic complex Ginzburg-Landau dynamics.

Abstract

We consider the numerical approximation of the stochastic complex Ginzburg-Landau equation with additive noise on the one dimensional torus. The complex nature of the equation means that many of the standard approaches developed for stochastic partial differential equations can not be directly applied. We use an energy approach to prove an existence and uniqueness result as well to obtain moment bounds on the stochastic PDE before introducing our numerical discretization. For such a well studied deterministic equation it is perhaps surprising that its numerical approximation in the stochastic setting has not been considered before. Our method is based on a spectral discretization in space and a Lie-Trotter splitting method in time. We obtain moment bounds for the numerical method before proving our main result: strong convergence on a set of arbitrarily large probability. From this we obtain a result on convergence in probability. We conclude with some numerical experiments that illustrate the effectiveness of our method.

Paper Structure

This paper contains 13 sections, 21 theorems, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and stochastic convolution
  3. Stochastic convolution
  4. Existence, uniqueness, and moment bounds of the SCGLE
  5. The non-linear term
  6. A priori estimates
  7. Existence and uniqueness results
  8. Numerical discretization
  9. Numerical method
  10. Moment bounds
  11. Convergence results
  12. Numerical results
  13. Space-time convergence

Key Result

Lemma 2.1

(i) For $u\in \dot{H}^{\beta}$ and $\beta\leq\alpha$ there is a constant $C_{\alpha,\beta,T}>0$ such that (ii) For $\gamma\geq 0$, there exists a constant $C_{T,\gamma}>0$ such that for $t \in (0,T]$ (iii) For $\gamma \in [0,1]$ and $t\in (0,T]$, there exists a constant $C_{T,\gamma}>0$ such that

Figures (3)

  • Figure 1: Defect turbulence setting with $\mu=-3$, $\nu=3$ and regular noise ($r=0$). (a) ${\rm Re} (u(x,t))$, (b) ${\rm Im} (u(x,t))$, (c) $|u(x,t)|$, and (d) the phase of the solution.
  • Figure 2: Convergence plot for the stable case $\mu=\nu=1$. (a) With regular noise ($r=0$) and (b) space-time white noise ($r=-\tfrac{1}{2}$)
  • Figure 3: Convergence plot for the defect turbulence case $\mu=-3$ and $\nu=3$. (a) With regular noise ($r=0$) and (b) with space-time white noise ($r=-\tfrac{1}{2}$).

Theorems & Definitions (42)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 32 more