An adaptive discontinuous Galerkin method for the simulation of hurricane storm surge

Abstract

Numerical simulations based on solving the 2D shallow water equations using a discontinuous Galerkin (DG) discretisation have evolved to be a viable tool for many geophysical applications. In the context of flood modelling, however, they have not yet been methodologically studied to a large extent. Systematic model testing is non-trivial as no comprehensive collection of numerical test cases exists to ensure the correctness of the implementation. Hence, the first part of this manuscript aims at collecting test cases from the literature that are generally useful for storm surge modellers and can be used to benchmark codes. On geographic scale, hurricane storm surge can be interpreted as a localised phenomenon making it ideally suited for adaptive mesh refinement (AMR). Past studies employing dynamic AMR have exclusively focused on nested meshes. For that reason, we have developed a DG storm surge model on a triangular and dynamically adaptive mesh. In order to increase computational efficiency, the refinement is driven by physics-based refinement indicators capturing major model sensitivities. Using idealised numerical test cases, we demonstrate the model's ability to correctly represent all source terms and reproduce known variability of coastal flooding with respect to hurricane characteristics such as size and approach speed. Finally, the adaptive mesh significantly reduces computing time with no effect on storm waves measured at discrete wave gauges just off the coast which shows the model's potential for use as a robust simulation tool for real-time predictions.

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

“…For solving the prescribed equations, we follow an approach similar to Beisiegel et al 7 The flux form from Equation (4) with the variables $$$ \mathbf{q} $$$, the flux $$$ \mathbf{F}\left(\mathbf{q}\right) $$$ and the source vector $$$ \mathbf{S}\left(\mathbf{q}\right) $$$ as previously described can be discretized by the DGM approach. The domain $$$ \Omega $$$ is decomposed into elements (either triangles or quadrilaterals in this study) $$$ {\Omega}_i $$$ such that $$$ \Omega =\cup {\Omega}_i $$$, approximating the variables $$$ {\mathbf{q}}_N\left(x,t\right)={\sum}_i{\mathbf{q}}_i(t){\psi}_i(x) $$$, where $$$ {\psi}_i(x) $$$ are Lagrange polynomials and $$$ {\mathbf{q}}_i(t) $$$ are (time‐dependant) coefficients (locally in each element).…”

“…6 AMATOS is a library that allows for adaptive mesh refinement. This is coupled with the so-called StormFlash library that has been used mostly for tsunami-modeling and shallow-water equations-based problems 7 but also has the ability to solve the Euler equations in a DGM context. The new scheme is implemented in StormFlash.…”

Shallow-Flow Velocity Predictions Using Discontinuous Galerkin Solutions

“…Alvarez-Vázquez et al (2008) applied a DG2 solver to simulate fish migration in coastal flows, also commenting on its ability to capture eddies. Beisiegel et al (2020) applied a DG2 solver to simulate hurricane-induced flow circulation in coastal areas, showing that it can replicate the asymmetrical patterns of the recirculation eddies extracted from a 3D Navier-Stokes computational model. Still, a dedicated study is needed to understand the extent to which a DG2 solver can reliably predict shallow-flow velocity fields at sub-meter grid resolution, which are featured by quasi-steady flow phenomena in and around hydraulic structures.…”

Probabilistic Characterization of Extreme Storm Surges Induced by Tropical Cyclones

“…(2018) used ROMS forced with meteorological forcing obtained from the U.S. National Centers for Environmental Prediction Climate Forecast System Reanalysis to simulate SSL along the Australian coastline. The process‐driven hydrodynamic models contribute to the understanding of physical mechanisms of the storm surge process and can predict storm surges accurately (Dietrich et al., 2011; Ramos‐Valle et al., 2020); however, they are often numerically expensive and may fail to fully resolve bathymetric and geometric features due to limited details in the available data sets or the resolution of computational grids, thereby affecting the applicability and accuracy of the SSL simulation (Arns et al., 2020; Beisiegel et al., 2020; Cyriac et al., 2018; Fernández‐Montblanc et al., 2019; Zou et al., 2013).…”