An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

• Published in: Journal of computational physics 2019-10, Vol.394, p.385-416
• Main Authors: Escalante, C., Dumbser, M., Castro, M.J.
• Format: Article
• Language: English
• Subjects: ADER discontinuous Galerkin schemes; Algorithms; Breaking waves; Computational fluid dynamics; Computational physics; Conservation laws; Divergence; Efficient parallel implementation on GPU; Eigenvalues; Exact solutions; Free surfaces; Galerkin method; Hydrostatic pressure; Hyperbolic reformulation; Magnetohydrodynamics; Mathematical models; Non-hydrostatic shallow water flows; Path-conservative finite volume methods; Riemann solver; Robustness (mathematics); Solitary waves; Velocity distribution; Water waves; Wave breaking; Wave propagation; Wave runup
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.05.035

In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by Sainte-Marie et al. in [1]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE. We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with a posteriori subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm. •New family of hyperbolic reformulations of depth-averaged models for nonlinear dispersive water waves.•Governing PDE system fulfills an extra energy conservation law (convex extension).•Discretization with arbitrary high order accurate discontinuous Galerkin finite element schemes.•The new approach is computationally efficient and allows large explicit time steps.•Straightforwa