We develop structure-preserving numerical methods for the Serre-Green-Naghdi equations, a model for weakly dispersive free-surface waves. We consider both the classical form, requiring the inversion of a non-linear elliptic operator, and a hyperbolic approximation of the equations, allowing fully explicit time stepping. Systems for both flat and variable topography are studied. Our novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. We provide a theoretical setting allowing us to construct schemes of any kind (finite difference, finite element, discontinuous Galerkin, spectral, etc.) as long as summation-by-parts operators are available in the chosen setting. Energy-stable variants are proposed by adding a consistent high-order artificial viscosity term. The proposed methods are validated through a large set of benchmarks to verify all the theoretical properties. Whenever possible, comparisons with exact, reference numerical, or experimental data are carried out. The impressive advantage of structure preservation, and in particular energy preservation, to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.

翻译：本文针对弱色散自由表面波模型——Serre-Green-Naghdi方程，发展了结构保持数值方法。我们同时研究了经典形式（需对非线性椭圆算子求逆）和方程的双曲近似形式（允许完全显式时间步进），并分析了平坦与变化地形下的系统。我们提出的新型数值方法能同时守恒总水质量和总能量。此外，针对原始Serre-Green-Naghdi方程的方法在平坦地形下还能守恒总动量。对于变化地形，所有提出的方法对静水稳态均满足良好平衡性。我们建立了理论框架，只要在所选设定中存在分部求和算子，即可构建任意类型（有限差分、有限元、间断Galerkin、谱方法等）的格式。通过添加一致的高阶人工粘性项，提出了能量稳定变体。通过大量基准测试验证了所提方法的所有理论特性，并在可能情况下与精确解、参考数值解或实验数据进行了对比。多个测试表明，结构保持（特别是能量保持）特性在极粗网格上精确解析色散波传播方面具有显著优势。