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

Abstract

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.

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

“…Due to their accuracy and compact stencil in the future we also plan to use P N P M schemes with a posteriori subcell finite volume limiter in the context of hyperbolic reformulations of nonlinear dispersive systems and wave propagation problems, see e.g. [11,12,34,54,58].…”

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

“…The derivation from the compressible Euler equations presented in [3] can also be used to establish a connection between the square of the sound speed in the compressible medium and the penalty parameter λ used in the model ( 1)- (5). However, the model proposed in [3] is non-conservative even for flat bottom (see also [51,52]). In the rest of this paper we sometimes refer to the model ( 1)-( 5) also as the Favrie-Gavrilyuk (FG) model, since it is the extension of [56] to the case of mildly varying bottom.…”

“…Synolakis [95] carried out laboratory experiments for solitary incident waves to study propagation, breaking and run-up over a planar beach with a slope 1 : 19.85. Since then, many researchers used his data to validate numerical models, as in [3,[51][52][53]68,90,91], among others. Accordingly, this test case is here used to assess the capability of the proposed methodology to describe shoreline motions and wave breaking when it occurs.…”

“…The idea goes back to the seminal work of Cattaneo [25], who proposed a hyperbolic reformulation of the parabolic heat equation. More recent work on the topic also regards the hyperbolic reformulation of advection-diffusion equations [80,84,85,99], of the compressible Navier-Stokes equations [17,47,88] as well as hyperbolic reformulations of nonlinear dispersive systems [3,4,26,51,52,63,64,79,90]. We would like to point out a particularity of the hyperbolic dispersive system proposed in [3], since it can be derived from the depth averaged compressible Euler equations, while most depth-averaged shallow water systems are usually derived from the governing equations of an incompressible fluid.…”

Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

“…Classic dispersive systems, and in particular DAE and SGN, may be written as a non-hydrostatic system. Readers can refer to [32] for the link between the two approaches. Non-hydrostatic formulation has several advantages from the numerical point of view, especially regarding boundary conditions.…”