Abstract
This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.
Highlights
In recent decades, due to the occurrence of a high number of coastal catastrophes, partly increased by the rise in sea level, the importance of research on wave propagation mechanisms in coastal areas has increased.Traditionally, depth-integrated models based on the Boussinesq-type equations (BTEs) have been used to model wave propagation, but the presence of the high order partial derivative terms contained in BTEs makes discretization difficult, tends to provoke numerical instabilities, and has a non-negligible computational cost
Depth-integrated models based on the Boussinesq-type equations (BTEs) have been used to model wave propagation, but the presence of the high order partial derivative terms contained in BTEs makes discretization difficult, tends to provoke numerical instabilities, and has a non-negligible computational cost
The capabilities of the non-hydrostatic models for water waves with a single layer or multiple layers in the vertical direction has been investigated by Stelling and Zijlema [3]; Zijlema et al [4]; Zijlema and Stelling [5]; Zijlema et al [6], among others
Summary
Due to the occurrence of a high number of coastal catastrophes, partly increased by the rise in sea level, the importance of research on wave propagation mechanisms in coastal areas has increased. Calvo & Rosman [13] presented a depth-integrated non-hydrostatic finite element model for wave propagation based in a continuous Galerkin finite element method. A discontinuous Galerkin method for one-dimensional nonhydrostatic depth-integrated flows was presented by Jeschke et al [17] Their model needs a quadratic vertical pressure profile and a carefully chosen scalar parameter in case of non-constant bathymetry. A new two-dimensional depth-integrated non-hydrostatic model capable of simulating wave propagation, breaking and runup in a finite element framework is constructed using a combination of continuous and discontinuous Galerkin methods. The second important novelty of this work is the development of a new slope limiter for quadrilateral elements based on the dissipative interface of Rosman [18], previously used in a continuous Galerkin depth-integrated shallow water model. Due to the linearization of the vertical momentum equation, it is well known that the depth-integrated, non-hydrostatic models are only applicable to the intermediate water depth and for weakly nonlinear cases [21]
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