[Paper Review] Mass and energy conservative high order diagonally implicit Runge--Kutta schemes for nonlinear Schrödinger equation in one and two dimensions

Ziyuan Liu, Hong Zhang+2 more|arXiv (Cornell University)|Oct 30, 2019

Numerical methods for differential equations16 references4 citations

TL;DR

This paper proposes high-order diagonally implicit Runge-Kutta (DIRK) schemes for the nonlinear Schrödinger equation in one and two dimensions, using the invariant energy quadratization (IEQ) approach to achieve both mass and energy conservation. The method combines Fourier pseudospectral spatial discretization with high-order DIRK time integrators, demonstrating up to fifth-order convergence, strict conservation of mass and energy (errors <10⁻¹⁴), and robust long-time stability in numerical experiments.

ABSTRACT

We present and analyze a series of conservative diagonally implicit Runge--Kutta schemes for the nonlinear Schrödiner equation. With the application of the newly developed invariant energy quadratization approach, these schemes possess not only high accuracy , high order convergence (up to fifth order) and efficiency due to diagonally implicity but also mass and energy conservative properties. Both theoretical analysis and numerical experiments of one- and two-dimensional dynamics are carried out to verify the invariant conservative properties, convergence orders and longtime simulation stability.

Motivation & Objective

• To develop high-order, energy- and mass-conserving time integrators for the nonlinear Schrödinger equation (NLS) in 1D and 2D.
• To address the challenge of preserving both mass and energy in long-time simulations of NLS with nonlinear terms.
• To combine the invariant energy quadratization (IEQ) approach with diagonally implicit Runge-Kutta (DIRK) schemes to achieve arbitrary high-order accuracy while preserving conservation laws.
• To rigorously prove the conservative properties of the semi-discrete and fully discrete systems.
• To validate the method’s performance through numerical experiments on convergence, conservation, and singularity capture.

Proposed method

• Apply the invariant energy quadratization (IEQ) approach to reformulate the NLS equation into an equivalent system with quadratic energy structure, enabling semi-explicit treatment of nonlinear terms.
• Discretize the spatial derivatives using the Fourier pseudospectral method, which provides spectral accuracy and preserves the structure of the PDE in space.
• Employ diagonally implicit Runge-Kutta (DIRK) schemes of order 2 to 5 in time, ensuring high-order accuracy and A-stability while maintaining diagonal structure for efficient solution.
• Prove that the semi-discrete and fully discrete systems conserve mass and energy under the condition that the stage values satisfy a specific algebraic constraint (M=0).
• Use the IEQ reformulation to transform the original NLS into a system with a modified energy that is preserved by the time integrator.
• Implement the schemes for both 1D and 2D NLS equations, including periodic boundary conditions and complex-valued solutions.

Experimental results

Research questions

• RQ1 Can high-order diagonally implicit Runge-Kutta schemes be constructed to preserve both mass and energy in the time integration of the nonlinear Schrödinger equation?
• RQ2 Does the invariant energy quadratization (IEQ) approach enable the construction of conservative, high-order time integrators for NLS with arbitrary polynomial nonlinearities?
• RQ3 What is the convergence order of the proposed IEQ-DIRK schemes in time, and how do they perform in long-time simulations?
• RQ4 Can the IEQ-DIRK schemes accurately capture singularities and sharp gradients in the solution of the NLS equation?
• RQ5 How do the mass and energy errors evolve over long-time simulations, and can they be kept at machine precision levels?

Key findings

• The proposed IEQ-DIRK schemes achieve up to fifth-order convergence in time, as confirmed by log-log plots of $L^2$ errors against time step sizes.
• Mass and energy errors are numerically negligible (below $10^{-14}$) across all tested schemes (2nd to 5th order), confirming perfect conservation within machine precision.
• The schemes maintain long-time stability, as demonstrated by simulations over $T=1000$ with $ riangle t = 0.01$, where mass and energy errors remain bounded and at machine level.
• The method accurately captures the dynamics of singular solutions in 2D NLS, including rapid changes in $|u_t|$, with mass and energy errors remaining well-controlled.
• The numerical results for the 1D traveling wave solution confirm that both IEQ-DIRK(2,2) and IEQ-DIRK(4,4) reproduce the exact soliton profile with high accuracy.
• The theoretical proof shows that the condition $M = oldsymbol{0}$ is both necessary and sufficient for the IEQ-DIRK schemes to preserve mass and energy, establishing a key analytical foundation.