Low regularity symplectic schemes for stochastic NLS

1. Problem Statement

The paper addresses the numerical approximation of the Stochastic Nonlinear Schrödinger Equation (SNLSE) in one dimension, driven by space-time white noise (specifically Stratonovich noise). The equation is given by:
$$i\partial_t u + \partial_x^2 u = \lambda |u|^2 u + \kappa u \circ \Phi \xi(t, x)$$
where $u$ is defined on a 1D torus $\mathbb{T}$, $\lambda$ and $\kappa$ are constants, and $\Phi \xi$ represents the noise.

Key Challenges:

• Low Regularity: Standard numerical methods (like traditional Runge-Kutta or splitting methods) often require high regularity of the initial data (e.g., $H^s$ with $s > 3/2$) to achieve convergence. This is due to the need for Taylor expansions of oscillatory integrals in the Duhamel formulation, which fail for rough initial data.
• Symplecticity: In Hamiltonian systems, preserving geometric structures (like mass and symplectic form) is crucial for long-time stability. However, constructing symplectic schemes for stochastic PDEs is difficult, especially when dealing with low regularity.
• Stochastic Integrability: Pathwise analysis of stochastic iterated integrals (arising from Duhamel's formula) is problematic. Previous work showed that without specific cancellations, strong convergence is limited by the lack of pathwise integrability of stochastic terms.
• Implicitness vs. Stability: Symplectic schemes for nonlinear terms are inherently implicit. In the stochastic setting, the stability of implicit schemes often depends on a time-step size that is random and inversely proportional to the noise norm, making global stability theoretically impossible to guarantee almost surely.

2. Methodology

The authors develop a class of Symplectic Resonance-Based Runge-Kutta (RK) schemes. The methodology combines techniques from deterministic low-regularity integrators with stochastic analysis.

A. Resonance-Based Kernel Approximation

Instead of using standard Taylor expansions for the oscillatory kernels in the Duhamel formulation, the authors employ a resonance-based approach (building on [17]).

• They decompose the oscillatory kernel $e^{-2iskk_1 + 2isk_2k_3}$ into dominant and lower parts.
• They use a specific approximation $K(s; k, k_1, k_2, k_3) \approx e^{-2iskk_1} + e^{2isk_2k_3} - 1$.
• This allows for the exact integration of dominant oscillatory terms while treating lower-order terms via polynomial interpolation. This significantly reduces the regularity requirements on the initial data.

B. Stochastic Discretization and Cancellation

A critical innovation is the treatment of the stochastic convolution terms:

• Double Iterated Integrals: By applying a symmetric midpoint rule to the Duhamel iteration, the authors show that the second iteration of the stochastic integral (order $t$) cancels out up to order $t^2$.
• Specifically, they utilize the property that $\int_0^s \circ dW(r) + \int_s^t \circ dW(r)$ combined with Stratonovich calculus leads to a vanishing term for the double integral in the first-order scheme.
• This allows the construction of a first-order scheme that avoids the problematic double stochastic integral, relying only on single integrals of order $\sqrt{t}$.

C. Symplectic Structure Preservation

The scheme is constructed to preserve the symplectic form $\omega = \int d\bar{u} \wedge du dx$.

• The authors define a general $S$-stage implicit RK scheme with coefficients satisfying specific algebraic constraints (Hypothesis 4.2).
• They prove that under these constraints and assuming the noise operator $\Phi$ is self-adjoint and symmetric in Fourier space, the scheme preserves the mass ($L^2$ norm) and the symplectic structure (in the sense of averaged energy).

D. The Stochastic Resonance Midpoint Rule

As a concrete example, they derive the Stochastic Resonance Midpoint Rule:
$$u_{n+1} = e^{it\partial_x^2} u_n + e^{it\partial_x^2} \left( t F_0(t; 1; u_{n+1/2}) + \sqrt{t} P_0(t; 1; u_{n+1/2}) \right)$$
where $u_{n+1/2}$ is an implicit midpoint approximation involving both deterministic and stochastic terms.

3. Key Contributions

1. New Class of Schemes: Introduction of a general class of symplectic, resonance-based Runge-Kutta schemes for SPDEs that handle low regularity initial data.
2. Low Regularity Convergence: The schemes achieve convergence for initial data with regularity lower than what is required by traditional methods (improving upon the $H^1$ barrier for strong convergence in some contexts).
3. Symplecticity in Stochastic Setting: Proof that the proposed schemes preserve the symplectic structure and mass, even in the presence of multiplicative Stratonovich noise.
4. Cancellation of Higher-Order Stochastic Terms: Demonstration that using a symmetric midpoint iteration causes the second-order stochastic iterated integral to vanish, simplifying the scheme and improving convergence properties.
5. Theoretical vs. Practical Stability Analysis:
   • Theory: The authors prove that the fixed-point iteration required for the implicit scheme is only stable if the time step $t$ satisfies a restrictive random bound (dependent on the noise norm). Consequently, global strong convergence cannot be proven almost surely due to the possibility of the scheme becoming ill-posed.
   • Practice: Numerical experiments show that despite the theoretical instability, the schemes are stable and convergent in practice for long time intervals, likely due to finite computational resources preventing the "rare" blow-up events.

4. Results

Theoretical Results

• Local Error: The local error of the midpoint rule is proven to be of order $O(t^{3/2})$ almost surely.
• Conservation: The scheme exactly conserves mass and preserves the symplectic form (Propositions 5.3 and 5.4).
• Stability Condition: The existence of a unique solution via fixed-point iteration requires $t < C / (1 + \|\Phi W\|)$, where $W$ is the noise. This implies the scheme is not well-posed with probability 1 for any fixed $t$.

Numerical Experiments

The authors compare their Resonance Midpoint Scheme against the Stochastic Crank-Nicolson method (a standard benchmark).

• Spatial Resolution: The Resonance scheme performs significantly better on refined spatial meshes (e.g., $2^{13}$ modes). Crank-Nicolson fails to converge or requires extremely small time steps on such meshes.
• Time Steps: The Resonance scheme allows for much larger time steps ($10^{-2}$) while maintaining stability on fine grids, whereas Crank-Nicolson becomes unstable.
• Convergence Order:
  • In low regularity settings, the Resonance scheme maintains a convergence order of $O(t)$, while Crank-Nicolson degrades significantly.
  • In high regularity settings, both schemes show similar convergence orders, though Crank-Nicolson has slightly better mass conservation at coarse resolutions.
• Long-Time Evolution: The Resonance scheme successfully simulates the SNLSE over long time frames ($T=1000$) with high spatial resolution, capturing dispersive effects and energy cascades that are computationally infeasible with Crank-Nicolson due to step-size restrictions.

5. Significance

This work bridges the gap between geometric integration (symplecticity) and low-regularity numerical analysis for stochastic PDEs.

• Overcoming Regularity Barriers: By avoiding Taylor expansions of oscillatory kernels, the method opens the door to simulating SNLSE with rough initial data, which is relevant for physical systems with turbulent or noisy initial conditions.
• Long-Time Simulation: The ability to use large time steps on fine spatial grids makes these schemes ideal for studying long-time phenomena like energy cascading and ergodicity in dispersive systems, areas where traditional methods are computationally prohibitive.
• Theoretical Insight: The paper provides a rigorous analysis of the tension between the theoretical requirement for small random time steps (for stability) and the practical observation of stability, highlighting the role of finite computational resources in numerical SPDE simulations.

In summary, the paper proposes a robust, symplectic numerical framework for stochastic dispersive equations that outperforms traditional methods in high-resolution, long-time simulations, despite theoretical limitations regarding almost-sure global stability.