Shallow water equations for large bathymetry variations

In this study, we propose an improved version of the nonlinear shallow water (or Saint-Venant) equations. This new model is designed to take into account the effects resulting from the large spacial and/or temporal variations of the seabed. The model is derived from a variational principle by choosing the appropriate shallow water ansatz and imposing suitable constraints. Thus, the derivation procedure does not explicitly involve any small parameter.Comment: 7 pages. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Shallow water equations for large bathymetry variations

In this study, we propose an improved version of the nonlinear shallow water (or Saint-Venant) equations. This new model is designed to take into account the effects resulting from the large spacial and/or temporal variations of the seabed. The model is derived from a variational principle by choosing the appropriate shallow water ansatz and imposing suitable constraints. Thus, the derivation procedure does not explicitly involve any small parameter.Comment: 7 pages. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Near-coast tsunami waveguiding: phenomenon and simulations

In this paper we show that shallow, elongated parts in a sloping bottom toward the coast will act as a waveguide and lead to large enhanced wave amplification for tsunami waves. Since this is even the case for narrow shallow regions, near-coast tsunami waveguiding may contribute to an explanation that tsunami heights and coastal effects as observed in reality show such high variability along the coastline. For accurate simulations, the complicated flow near the waveguide has to be resolved accurately, and grids that are too coarse will greatly underestimate the effects. We will present some results of extensive simulations using shallow water and a linear dispersive Variational Boussinesq model.\u

Near coast tsunami waveguiding: simulations for various wave models

Shallow parts in a sloping bottom toward the coast can be expected to act as a waveguide, in partial analogy with optical waveguiding. We will present numerical simulations that convincingly show that large enhanced wave amplification happens for tsunami waves in certain geometries and bathymetries. Since this is even the case for shallow regions that have cross sections of the order of badly resolved numerical scales, this phenomenon may at least partly explain that tsunami heights and coastal effects as observed in reality show such high variability along the coastline. This report, following [1], supports a more concise publication that will be published soon [2]. In this report we will provide detailed results of extensive simulations using various wave models and different gridsizes. We will show results obtained with the commonly used Shallow Water Equations and with a more accurate dispersive wave model. For the latter simulations we use a recently derived Variational Boussinesq model. We will also show that relatively small gridsizes are needed to capture the transversal flow near the waveguide; on grids that are too coarse, the enhanced amplification will not be observable. This may provide a partial explanation that spatial variability due to relatively shallow bottom variations will not be present in most simulations

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.Comment: Revised version, to appear in Annales de l'Institut Henri Poincar\'

Effective coastal boundary conditions for dispersive tsunami propagation

We aim to improve the techniques to predict tsunami wave heights along the coast. The modeling of tsunamis with the shallow water equations has been very successful, but is somewhat simplistic because wave dispersion is neglected. To bypass this shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [J. Fluid Mech. 657, 36--63, 2010]. Another shortcoming is that the complicated interactions between incoming and reflected waves near the shore are usually simplified by a fixed wall boundary condition at a certain shallow depth contour. To alleviate this shortcoming, we explore and present in one spatial dimension a so-called effective boundary condition (EBC). From the deep ocean to the seaward boundary, i.e., the simulation area, we model wave propagation numerically. Given the measurements of the incoming wave at the seaward boundary, we model the wave dynamics towards the shoreline analytically, based on shallow water theory and the Wentzel-Kramer-Brillouin (WKB) approximation, as well as extensions to the dispersive, Boussinesq model. The reflected wave is then influxed back into the simulation area using the EBC. The coupling between the two areas, one done numerically and one analytically, via the EBC is handled using variational principles, to preserve the overall energy in both areas. We verify and validate our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a non-destructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well-balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme -- even on unstructured grids -- and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, super-linear convergence, mass-conservation, well-balancedness, and stability are verified