Abstract
The shallow water equations provide the basic modeling equations for a number of coastal flooding hazards, such as tsunamis and storm surge. In realistic scenarios, there are often structures important to these flows that have a large extent but small width, including sea walls, berms, and harbor barriers. Explicit time stepping schemes, most often used for the shallow water equations, can then suffer from time step restrictions due to the CFL condition. In this article, we introduce an approach that side-steps these issues by allowing a barrier to have zero-width and to cut a cell arbitrarily without suffering from CFL restrictions. This is done by supplementing existing Riemann solvers and leveraging $h$-box and large time stepping methods. These methods preserve the properties of the Riemann solver and add negligible cost to the original solvers.
Highlights
Coastal flooding events constitute a major threat to communities along the coastlines throughout the world
We present a method for modeling barriers, such as sea walls, without fully resolving them in the context of the shallow water equations defined in one dimension by ht +x = 0,t
We present a set of extensions to an existing Riemann solver [15] and numerical techniques for the representation of a barrier that acts as both a reflecting wall boundary condition until it is overtopped and acts as a flux-limiting boundary
Summary
Coastal flooding events constitute a major threat to communities along the coastlines throughout the world. We again consider a barrier that is not aligned with the grid, including all forms of overtopping At this point the approach for specifying the ghost fluid is currently unknown, as the states needed are ambiguous, so we instead supplement the h-box approach with a large time-stepping (LTS) method. We will redistribute these waves, similar to how other similar methods redistribute fluxes, such that they will impact the original cells on either side of the wall If carefully done, this will maintain conservation and ensure that water flows from one side of the barrier to the other as determined by the given Riemann solver. Note that these averages will preserve the lake at rest case as long as a well-balanced Riemann solver is being used
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