An extension of the Boussinesq-type models to weakly compressible flows is derived in the fully nonlinear case. The dispersive properties are consistent with the linear theory of compressible fluids at the long-wave limit. The particular case of a vanishing Mach number gives a quasi-incompressible model, intended for coastal wave simulations, which is a hyperbolic version of the Serre–Green–Naghdi equations, with a new treatment of the bathymetric terms. Both the compressible and quasi-incompressible models are hyperbolic four-equation models on an arbitrary bathymetry, with an exact equation of energy conservation. In addition, these models are extended to hyperbolic and fully nonlinear five-equation versions with improved dispersive properties. A remarkable property of the quasi-incompressible model with improved dispersive properties is that it is possible to decrease significantly the sound velocity, and thus the computational time, with the same accuracy or even slightly better. The numerical results show good agreement of the quasi-incompressible model with experimental data and the capability of the compressible model to calculate the decrease of tsunami velocity due to compressible effects.

Boussinesq 型模型对弱可压缩流的扩展是在完全非线性的情况下导出的。色散特性与可压缩流体在长波极限处的线性理论一致。马赫数消失的特殊情况给出了一个准不可压缩模型，用于海岸波模拟，它是 Serre-Green-Naghdi 方程的双曲线版本，对测深项进行了新的处理。可压缩模型和准不可压缩模型都是任意测深上的双曲四方程模型，具有精确的能量守恒方程。此外，这些模型还扩展到具有改进色散特性的双曲线和完全非线性五方程版本。具有改进的色散特性的准不可压缩模型的一个显着特性是可以显着降低声速，从而以相同的精度甚至略好一点的精度降低计算时间。数值结果表明，准不可压缩模型与实验数据吻合良好，并且可压缩模型能够计算由于可压缩效应引起的海啸速度下降。