Conservative Integrators for a Toy Model of Weak Turbulence

TL;DR

This paper develops and analyzes numerical schemes that conserve invariants for a toy model of weak turbulence, demonstrating their effectiveness in long-term energy transfer simulations related to the 2D nonlinear Schrödinger equation.

Contribution

It introduces and proves convergence of invariant-preserving numerical schemes for a toy model of weak turbulence, improving long-term simulation accuracy.

Findings

Schemes compare favorably to Trapezoidal Rule and Runge-Kutta methods.

Invariant preservation is crucial for accurate long-term energy transfer modeling.

Convergence is established in certain cases.

Abstract

Weak turbulence is a phenomenon by which a system generically transfers energy from low to high wave numbers, while persisting for all finite time. It has been conjectured by Bourgain that the 2D defocusing nonlinear Schr\"odinger equation (NLS) on the torus has this dynamic, and several analytical and numerical studies have worked towards addressing this point. In the process of studying the conjecture, Colliander, Keel, Staffilani, Takaoka, and Tao introduced a "toy model" dynamical system as an approximation of NLS, which has been subsequently studied numerically. In this work, we formulate and examine several numerical schemes for integrating this model equation. The model has two invariants, and our schemes aim to conserve at least one of them. We prove convergence in some cases, and our numerical studies show that the schemes compare favorably to others, such as Trapezoidal Rule and fixed step fourth order Runge-Kutta. The preservation of the invariants is particularly important in the study of weak turbulence as the energy transfer tends to occur on long time scales.