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Splitting methods for the Gross-Pitaevskii equation on the full space and vortex nucleation

Quentin Chauleur, Gaspard Kemlin

Abstract

We prove the convergence in Zhidkov spaces of the first-order Lie-Trotter and the second-order Strang splitting schemes for the time integration of the Gross-Pitaesvkii equation with a time-dependent potential and non-zero boundary conditions at infinity. We also show the conservation of the generalized mass and the near-preservation of the Ginzburg-Landau energy balance law. Numerical accuracy tests performed on a one-dimensional dark soliton corroborate our theoretical findings. We finally investigate the nucleation of quantum vortices in two experimentally relevant settings.

Splitting methods for the Gross-Pitaevskii equation on the full space and vortex nucleation

Abstract

We prove the convergence in Zhidkov spaces of the first-order Lie-Trotter and the second-order Strang splitting schemes for the time integration of the Gross-Pitaesvkii equation with a time-dependent potential and non-zero boundary conditions at infinity. We also show the conservation of the generalized mass and the near-preservation of the Ginzburg-Landau energy balance law. Numerical accuracy tests performed on a one-dimensional dark soliton corroborate our theoretical findings. We finally investigate the nucleation of quantum vortices in two experimentally relevant settings.
Paper Structure (24 sections, 22 theorems, 221 equations, 6 figures, 1 table)

This paper contains 24 sections, 22 theorems, 221 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Functional framework
  4. Splitting methods
  5. A related equation and associated schemes
  6. Continuity properties
  7. Propagation of high regularity
  8. Stability estimates
  9. Local error and convergence
  10. A primer on Lie derivatives
  11. Commutator estimates
  12. Taylor formulas and local error
  13. Non-autonomous systems
  14. Convergence estimates
  15. Proof of Proposition \ref{['prop:convergence_splitting_v']}.
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $1\leq d \leq 3$ and $T>0$. Assume that $u_0=\phi + v_0$ with $v_0 \in H^4(\mathbb{R}^d)$, $V~\in~\mathcal{C}^1(\left[0,T\right],H^4(\mathbb{R}^d))$ and $\phi$ satisfying eq:requirements_phi. We denote by $u$ the unique global solution to eq:GP with initial data $u_0$. (Lie splitting). There exi where $\tau_{\mathcal{L}}$, $C_{\mathcal{L}}$ depend on $d,T,\|v_0\|_{H^4}, \|\phi\|_{X^4},\|V\|_{\

Figures (6)

  • Figure 1: Exact (u_ref) and approximate (u) solutions to \ref{['eq:GP']} in one dimension, obtained with the Strang splitting scheme, at time $t=0$ (top) and $t=2$ (bottom). The spatial domain has been restricted to $[-10,10]$ for plot clarity.
  • Figure 2: Visual near-preservation of the energy (top) and preservation of the mass -- up to numerical accuracy -- for both the Lie (left) and Strang (right) splitting schemes.
  • Figure 3: Case where $u_0(x) = \phi_c(x)$ is given by a dark soliton. (Left) Convergence in $\tau$ of both Lie and Strang splitting schemes. The error is computed as $\|u^N - u(T)\|_{X^2}$, where $u^N$ is the approximation at time $T$ and $u(T)$ is the exact solution. (Right) Super-convergence in $\tau$ of the energy for both Lie and Strang splitting schemes. The error is computed as $|\mathcal{E}(u^N) - \mathcal{E}(u(T))|$.
  • Figure 4: Case where $u_0(x) = \phi_c(x) - \frac{1}{2}\exp(-x^2)$. (Left) Convergence in $\tau$ of both Lie and Strang splitting schemes. The error is computed as $\|u^N - u(T)\|_{X^2}$, where $u^N$ is the approximation at time $T$ and $u(T)$ is the exact solution. (Right) Convergence in $\tau$ of the energy for both Lie and Strang splitting schemes. The error is computed as $|\mathcal{E}(u^N) - \mathcal{E}(u(T))|$. This time, we observe the expected order.
  • Figure 5: Numerical simulation by the Strang splitting scheme \ref{['eq:Strang_split_u']} of the solution $u$ to \ref{['eq:GP']} with a linearly moving Gaussian obstacle \ref{['eq:potential_linear']} (Case (i)) and parameters given in Table \ref{['table:parameters']}. (Left) density $|u|^2$ -- (Middle) phase -- (Right) Potential $V(t,\cdot)$, displayed at different times $t=0, 0.4, 0.8, 2.0$, with a zoom on the cone at $t=0.8$ (bottom line).
  • ...and 1 more figures

Theorems & Definitions (46)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Proposition 3.1
  • ...and 36 more