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

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