A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

• Published in: Journal of scientific computing Vol. 83; no. 3; p. 62
• Main Authors: Escalante, C., de Luna, T. Morales
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2020; Springer Nature B.V
• Subjects: Algorithms; Approximation; Atmospheric pressure; Boussinesq equations; Computational Mathematics and Numerical Analysis; Conservation laws; Dispersion; Divergence; Finite volume method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Ocean engineering; Theoretical; Velocity; Water waves; Non-hydrostatic shallow water flows; Well-balanced schemes; Boussinesq-type systems; Breaking waves; Path-conservative finite volume methods; Hyperbolic reformulation
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-020-01244-7

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.