[Paper Review] High order explicit symplectic integrators for the Discrete Non Linear Schrödinger equation

Jehan Boreux, Timotéo Carletti+1 more|arXiv (Cornell University)|Dec 15, 2010

Numerical methods for differential equations11 references3 citations

TL;DR

This paper presents a family of high-order explicit symplectic integrators for the Discrete Nonlinear Schrödinger Equation (DNLS), based on Yoshida's composition method, enabling long-term numerical simulations with exceptional energy and mass conservation. The method achieves orders up to 10 while remaining explicit and computationally efficient, outperforming non-symplectic and implicit schemes in accuracy and stability over large time spans.

ABSTRACT

We propose a family of reliable symplectic integrators adapted to the Discrete Non-Linear Schrödinger equation; based on an idea of Yoshida (H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A, 150, 5,6,7, (1990), pp. 262.) we can construct high order numerical schemes, that result to be explicit methods and thus very fast. The performances of the integrators are discussed, studied as functions of the integration time step and compared with some non symplectic methods.

Motivation & Objective

• To develop high-order explicit symplectic integrators for the Discrete Nonlinear Schrödinger Equation (DNLS), which lacks a separable Hamiltonian structure and thus resists standard symplectic methods.
• To overcome the limitations of implicit symplectic methods (e.g., Gauss-Radau, generating function-based schemes) that require costly Newton iterations and are unsuitable for large-scale DNLS simulations.
• To ensure long-term preservation of key physical invariants—energy, mass, and the conjugacy relation $\overline{p} = q$—critical for studying asymptotic dynamics like breathers and negative temperature regimes.
• To balance computational efficiency and accuracy by optimizing time step and integrator order, minimizing CPU cost while maintaining machine-precision energy conservation.
• To demonstrate that explicit high-order symplectic schemes, derived via Yoshida's composition technique, are viable and superior for DNLS despite the system's non-quasi-integrable nature.

Proposed method

• Adapt Yoshida's composition method to construct high-order symplectic integrators for the DNLS, using recursive composition of second-order symplectic maps to achieve orders 2, 4, 6, 8, 10.
• Define the Hamiltonian of the DNLS system as $ H = -i \sum_{l=1}^{N} \left[ \frac{(p_{l+1}-p_l)(q_{l+1}-q_l)}{h^2} - p_l^2 q_l^2 \right] $, which is not separable into kinetic and potential energy terms, precluding standard splitting methods.
• Construct the integrator maps $ \exp(L_A) $ and $ \exp(L_B) $, where $ L_A $ and $ L_B $ represent the linear and nonlinear parts of the Hamiltonian flow, respectively, using vector-matrix products for $ \exp(L_B) $.
• Implement the integrator using a composition of $ \exp(L_A) $, $ \exp(L_B) $, and their time-reversed counterparts with optimized coefficients $ (c_j, d_j) $, ensuring symplecticity and high-order accuracy.
• Optimize the scheme by using an improved version of Yoshida’s method ($ Y^{\text{opt}}_{2m} $) to reduce the number of composition steps, thereby decreasing CPU cost without sacrificing order.
• Leverage the Fourier-like structure of $ \exp(L_B) $ to accelerate computation via Fast Fourier Transform (FFT)-based methods, reducing $ \mathcal{O}(N^2) $ cost to $ \mathcal{O}(N \log N) $.

Experimental results

Research questions

• RQ1 Can high-order explicit symplectic integrators be constructed for the DNLS system, which lacks a separable Hamiltonian structure and thus resists standard symplectic splitting methods?
• RQ2 How do the energy and mass conservation properties of the proposed symplectic integrators compare to non-symplectic and implicit schemes over long integration times?
• RQ3 What is the trade-off between integrator order, time step size, and computational cost (CPU time) in achieving desired accuracy and energy preservation?
• RQ4 To what extent is the conjugacy relation $ \overline{p(t)} = q(t) $ preserved by the numerical scheme if it holds initially?
• RQ5 Can the proposed method achieve machine-precision energy conservation with large time steps, enabling efficient simulation of long-time asymptotic dynamics in DNLS?

Key findings

• The proposed high-order explicit symplectic integrators ($ Y_{2m} $) preserve the total energy with relative error below $ 10^{-15} $ when using time steps $ \tau \sim 10^{-3} $ for $ Y_4 $, and up to $ \tau \sim 10^{-2} $ for $ Y_6 $, demonstrating excellent long-term energy conservation.
• The mass integral $ I = \sum_l p_l q_l $ is preserved with near-machine precision, confirming the method's ability to maintain key physical invariants.
• The conjugacy relation $ \overline{p(t)} = q(t) $ is preserved numerically if it holds at $ t = 0 $, a critical property for modeling physical systems like Bose-Einstein condensates.
• The CPU time scales approximately as $ 1/\tau $ for fixed integrator order and as $ 3^m $ for fixed $ \tau $, indicating exponential growth with order $ 2m $, but the optimized $ Y^{\text{opt}}_{2m} $ versions reduce this cost by 1.5× ($ Y_6 $) and 2.2× ($ Y_8 $) compared to standard $ Y_{2m} $.
• For equivalent accuracy, $ Y_6(\tau = 10^{-2}) $ requires less CPU time than $ Y_4(\tau = 10^{-3}) $, and $ Y_8(\tau = 2 \times 10^{-2}) $ is more efficient than $ Y_6(\tau = 10^{-2}) $, showing that higher-order schemes can be more efficient at larger time steps.
• The modified Runge-Kutta 4 method is faster than $ Y_4 $ but slower than $ Y^{\text{opt}}_6 $, and the standard RK4 is significantly more expensive than $ Y_4 $, highlighting the superiority of the symplectic approach in terms of efficiency and accuracy.