High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization

TL;DR

This work addresses long-time accurate simulation of the nonlinear Schrödinger equation by combining arbitrarily high-order SBP spatial discretizations with a quadratic-preserving relaxation in time to conserve both mass and energy at the fully discrete level. It shows that standard Fourier and SBP-based schemes can be energy-conserving when the energy is formulated appropriately, and it introduces a scalable relaxation-based time integrator that avoids large nonlinear solves while preserving two nonlinear invariants. The paper provides extensive numerical evidence across soliton interactions, dispersive shocks, and a hyperbolized NLS, demonstrating high-order convergence, linear-in-time error growth with relaxation, and substantial performance gains. The approach is poised to improve long-time simulations in multi-dimensional settings and could extend to other systems with multiple conserved functionals.

Abstract

We propose a class of numerical methods for the nonlinear Schrödinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation per time step. We show that some existing spatial discretizations, including the popular Fourier spectral method, are in fact energy-conserving if one considers the appropriate form of the energy density. We develop a new relaxation-type approach for conserving multiple nonlinear functionals that is more efficient and robust for the NLS equation compared to the existing multiple-relaxation approach. The accuracy and efficiency of the new schemes is demonstrated on test problems for both the focusing and defocusing NLS.

Figures (10)

• Figure 1: Absolute changes of the discretized invariants of the nonlinear Schrödinger equation for Fourier collocation methods with $N \in \{64, 65\}$ nodes in the domain $[-35, 35]$ for the two-soliton setup. The semidiscretization is integrated in time with the fifth-order method of kennedy2019higher and time step size $\Delta t = 10^{-4}$.
• Figure 2: Absolute changes of the discretized invariants of the nonlinear Schrödinger equation for the two-soliton setup. The semidiscretizations are integrated in time with the fifth-order method of kennedy2019higher and time step size $\Delta t = 10^{-4}$. For all semidiscretizations, we use approximately 64 degrees of freedom with slight deviations depending on the polynomial degree $p$ for Galerkin methods.
• Figure 3: Spatial convergence study for the two-soliton setup. The semidiscretizations are integrated in time with the fifth-order method of kennedy2019higher and time step size $\Delta t = 5 \times 10^{-5}$. For finite difference methods, the "order" refers to the order of accuracy in the interior. For Galerkin methods, $p$ is the polynomial degree.
• Figure 4: Numerical results for the dispersive shock wave obtained with Fourier collocation methods and the fifth-order method of kennedy2019higher.
• Figure 5: Numerical results obtained for the two- ($\Delta t = 10^{-2}$) and three-soliton ($\Delta t = 10^{-3}$) setups using Fourier collocation methods with $N = 2^{10}$ nodes in the domain $[-35, 35]$ in space and the third-order method of ascher1997implicit in time.

Theorems & Definitions (10)

• definition 2.1
• definition 2.2
• definition 2.3
• theorem 2.4
• proof
• theorem 3.1
• remark 3.2
• proof : Proof of Theorem \ref{['thm:geodesic_relaxation']}
• theorem 5.1
• proof