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。我们同时研究了需要非线性椭圆算子求逆的经典形式，以及允许全显式时间步进的双曲近似方程。研究涵盖平坦地形与变化地形两种系统。我们提出的新型数值方法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守恒总动量。对于变化地形，所有提出方法对静止湖状态均满足良平衡性。我们建立了理论框架，允许在选定的数值设置中（只要存在分部求和算子）构建任意类型的格式（有限差分、有限元、间断伽辽金、谱方法等）。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。多个测试案例证明了结构保持（特别是能量保持）在极粗网格上精确解析色散波传播的显著优势。