Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation

2025/05/06 by Erwan Faou, Faou, Erwan, Georg Maierhofer +3 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #Control Systems and Identification #FOS: Mathematics #Image and Signal Denoising Methods #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2505.03271

openalex publication_date 2025/05/06 · openalex created_date 2025/10/07 · openalex updated_date 2026/07/30

Abstract

The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is known in the context of Hamiltonian PDEs. In this work we provide the first dimension-independent backward error analysis for a Runge-Kutta-type method, the midpoint rule, which shows the existence of a modified energy for this method when applied to nonlinear Schroedinger equations regardless of the level of spatial discretisation. We use this to establish long-time stability of the numerical flow for the midpoint rule.

Citations

• The Gronwall inequality
• Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity
• Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes
• Existence and stability of traveling waves for discrete nonlinear Schroedinger equations over long times
• Plane wave stability of the split-step Fourier method for the nonlinear\n Schr "odinger equation
• Hamiltonian interpolation of splitting approximations for nonlinear PDEs
• �ber die partiellen Differenzengleichungen der mathematischen Physik

Cited by

• Computing nonlinear Schrödinger equations with Hermite functions beyond harmonic traps

Related

• Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
• Global Error Control in the Runge-Kutta Solution of a Hamiltonian System using the RKQ Algorithm
• Resonances in long time integration of semi linear Hamiltonian PDEs
• Low-Rank Solvers for Energy-Conserving Hamiltonian Boundary Value Methods
• A posteriori error analysis for evolution nonlinear Schr "odinger\n equations up to the critical exponent