Long-term behaviour of symmetric partitioned linear multistep methods II. Invariants error analysis for some nonlinear dispersive wave models

TL;DR

This work analyzes the long-time accuracy of symmetric partitioned linear multistep methods for semidiscretized nonlinear dispersive PDEs, focusing on solitary-wave dynamics in the nonlinear Schrödinger equation and Bona–Smith Bona-Smith systems. By combining periodic extensions, Fourier pseudospectral spatial discretization, and invariant-based error analysis, the authors derive bounds on the growth of errors in mass, momentum, and energy (Hamiltonian) under symmetric, order-$r$ PLMMs, including Landau-type constants and conditions on starting values. The results show that, under suitable root conditions and symmetry, invariant errors remain $O(Δt^r)$ for times up to $O(Δt^{-1})$ (and up to $O(Δt^{-2})$ with higher-order starts in some cases), with further cancellations for smooth versus non-smooth components in Bona–Smith systems. Numerical experiments corroborate the theory, illustrating linear stability near the origin for symmetric schemes, bounded invariant errors for solitary waves, and the potential for resolution of localized structures into trains of solitary waves, which underscores the practical impact of symmetry and invariants-preserving time integrators in long-time simulations of nonlinear dispersive dynamics.

Abstract

In this paper, the use of partitioned linear multistep methods (PLMM) as time integrators for the numerical approximation of some partial differential equations (pdes) is studied. We consider the periodic initial-value problem of two nonlinear dispersive wave models as case studies. From the spatial discretization with pseudospectral methods, the theory developed for PLMMs by the authors in a previous companion paper is applied to analyze the time integration with PLMMs of the semidiscrete equations when approximating solitary wave solutions. The results are illustrated with some numerical experiments. In addition, a computational study is performed in an exploratory fashion to analyze the extension of the results to the approximation of more general localized solutions.

Figures (10)

• Figure 1: Error in the invariants against time with NSPLMM2 w.r.t. a solitary wave (\ref{['sw']}) of the NLS equation (\ref{['nls2']}) with $\sigma=1$ and parameters $a=\lambda_{0}^{2}=1, x_{0}=-400, \theta_{0}=\pi/4$. (a) Mass error. (b) Energy error. $\Delta x=1.25\times 10^{-1}$.
• Figure 2: Error in the invariants against time with NSPLMM2 w.r.t. a solitary wave (\ref{['sw']}) of the NLS equation (\ref{['nls2']}) with $\sigma=1$ and parameters $a=\lambda_{0}^{2}=1, x_{0}=-400, \theta_{0}=\pi/4$. (a) Mass error. (b) Energy error. $\Delta x=1.25\times 10^{-1}$.
• Figure 3: Error in the invariants against time w.r.t. a solitary wave (\ref{['sw']}) of the NLS equation (\ref{['nls2']}) with $\sigma=1$ and parameters $a=\lambda_{0}^{2}=1, x_{0}=-400, \theta_{0}=\pi/4$. (a), (b) Mass and energy error for NSNPLMM3. (c), (d) Mass and energy error for SPLMM2. $\Delta x=1.25\times 10^{-1}$.
• Figure 4: Error in the Hamiltonian against time w.r.t. a solitary wave of Boussinesq system (\ref{['bb']}) with $\theta^{2}=9/11, b=d=\frac{1}{2}(\theta^{2}-1/3), c=2/3-\theta^{2}, a=0$ . $\Delta x=1.25\times 10^{-1}$. (a) Dashed lines: NSNPLMM3; solid lines: SPLMM2. (b) Dashed lines: NSPLMM2; solid lines: SPLMM2.
• Figure 5: Error in the Hamiltonian against time w.r.t. a solitary wave of Boussinesq system (\ref{['bb']}) with $\theta^{2}=9/11, b=d=\frac{1}{2}(\theta^{2}-1/3), c=2/3-\theta^{2}, a=0$. SPLMM2 with starting values obtained from the implicit midpoint rule.

Theorems & Definitions (8)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• proof
• Remark 2.2
• Remark 2.3
• Theorem 2.3
• Remark 2.4