In this paper, we propose a novel first-order reformulation of the most well-known Boussinesq-type systems that are used in ocean engineering. This has the advantage of collecting in a general framework many of the well-known systems used for dispersive flows. Moreover, it avoids the use of high-order derivatives which are not easy to treat numerically, due to the large stencil usually needed. These first-order PDE dispersive systems are then approximated by a novel set of first-order hyperbolic equations. Our new hyperbolic approximation is based on a relaxed augmented system in which the divergence constraints of the velocity flow variables are coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressures. The most important advantage of this new hyperbolic formulation is that it can be easily discretized with explicit and high-order accurate numerical schemes for hyperbolic conservation laws. There is no longer need of solving implicitly some linear system as it is usually done in many classical approaches of Boussinesq-type models. Here a third-order finite volume scheme based on a CWENO reconstruction has been used. The scheme is well-balanced and can treat correctly wet–dry areas and emerging topographies. Several numerical tests, which include idealized academic benchmarks and laboratory experiments are proposed, showing the advantage, efficiency and accuracy of the technique proposed here.

在本文中，我们提出了一种最新颖的用于海洋工程的最著名的Boussinesq型系统的一阶重构。这具有在通用框架中收集许多用于分散流的众所周知的系统的优点。此外，由于通常需要较大的模版，因此避免了使用不易数值处理的高阶导数。这些一阶PDE色散系统然后由一组新颖的一阶双曲方程组近似。我们新的双曲近似法基于松弛的增广系统，在该系统中，速度变量的散度约束通过深度平均非静水压力的演化方程与其他守恒律耦合。这种新的双曲线公式的最重要的优点是，可以通过双曲线守恒定律的显式和高阶精确数值方案轻松地离散化该双曲线公式。不再需要隐式求解某些线性系统，因为通常在Boussinesq型模型的许多经典方法中都需要这样做。这里使用了基于CWENO重构的三阶有限体积方案。该方案平衡良好，可以正确处理干湿地区和新兴地形。提出了几种数值测试，其中包括理想化的学术基准和实验室实验，显示了此处提出的技术的优势，效率和准确性。不再需要隐式求解某些线性系统，因为通常在Boussinesq型模型的许多经典方法中都需要这样做。这里使用了基于CWENO重构的三阶有限体积方案。该方案平衡良好，可以正确处理干湿地区和新兴地形。提出了几种数值测试，其中包括理想化的学术基准和实验室实验，显示了此处提出的技术的优势，效率和准确性。不再需要隐式求解某些线性系统，因为通常在Boussinesq型模型的许多经典方法中都需要这样做。这里使用了基于CWENO重构的三阶有限体积方案。该方案平衡良好，可以正确处理干湿地区和新兴地形。提出了几种数值测试，其中包括理想化的学术基准和实验室实验，显示了此处提出的技术的优势，效率和准确性。