General Initial Value Problem for the Nonlinear Shallow Water Equations: Runup of Long Waves on Sloping Beaches and Bays

TL;DR

This paper introduces a new method for solving the nonlinear shallow water equations using the Carrier-Greenspan transformation, enabling more accurate modeling of tsunami runup on complex beaches with realistic initial conditions.

Contribution

It extends existing solutions by incorporating Taylor series approximations to handle complex initial conditions and bathymetries in the transformed space.

Findings

Enables modeling of tsunami runup on U-shaped beaches.

Allows verification of tsunami inundation models in 2-D.

Handles large initial velocities and complex initial conditions.

Abstract

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier-Greenspan transformation [G. Carrier and H. Greenspan, J. Fluid Mech. 01, 97 (1957)]. We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.