Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations

2025/12/18 by Hendrik Ranocha, Ranocha, Hendrik, David I. Ketcheson +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #65M12 #65M20 #65M60 #65M70 #FOS: Mathematics #Model Reduction and Neural Networks #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Numerical methods for differential equations #cs.NA #math.NA #msc:65M12 #msc:65M20 #msc:65M60 #msc:65M70

paper · pdf · doi:10.48550/arxiv.2512.16352

The reproducibility repository is available at https://github.com/ranocha/2025_BBM_KdV_NLS and https://zenodo.org/doi/10.5281/zenodo.17936837

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/08/01 · arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We propose and study a class of arbitrarily high-order numerical discretizations that preserve multiple invariants and are essentially explicit (they do not require the solution of any large systems of algebraic equations). In space, we use Fourier Galerkin methods, while in time we use a combination of orthogonal projection and relaxation. We prove and numerically demonstrate the conservation properties of the method by applying it to the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger (NLS) PDEs as well as a hyperbolic approximation of NLS. For each of these equations, the proposed schemes conserve mass, momentum, and energy up to numerical precision. We show that this conservation leads to reduced growth of numerical errors for long-term simulations.

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• A Unified Discrete Gradient-SAV Framework for Structure-Preserving Integration

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