Accurate Solution of the Nonlinear Schr\"{o}dinger Equation via Conservative Multiple-Relaxation ImEx Methods

TL;DR

This paper introduces a conservative, explicit numerical method for the nonlinear Schrödinger equation that preserves key invariants, offering improved accuracy and efficiency for long-time simulations of multi-soliton solutions.

Contribution

It develops a novel explicit discretization combining ImEx Runge-Kutta methods with relaxation and adaptive step sizing to conserve invariants in NLS simulations.

Findings

Mass-conserving method outperforms 2nd-order splitting in accuracy and efficiency.

Adaptive time stepping reduces computational cost significantly.

Full discretization conserves both mass and energy, improving long-term solution accuracy.

Abstract

The nonlinear Schr\"{o}dinger (NLS) equation possesses an infinite hierarchy of conserved densities and the numerical preservation of some of these quantities is critical for accurate long-time simulations, particularly for multi-soliton solutions. We propose an essentially explicit discretization that conserves one or two of these conserved quantities by combining higher-order Implicit-Explicit (ImEx) Runge-Kutta time integrators with the relaxation technique and adaptive step size control. We show through numerical tests that our mass-conserving method is much more efficient and accurate than the widely-used 2nd-order time-splitting pseudospectral approach. Compared to higher-order operator splitting, it gives similar results in general and significantly better results near the semi-classical limit. Furthermore, for some problems adaptive time stepping provides a dramatic reduction in cost without sacrificing accuracy. We also propose a full discretization that conserves both mass and energy by using a conservative finite element spatial discretization and multiple relaxation in time. Our results suggest that this method provides a qualitative improvement in long-time error growth for multi-soliton solutions.