Wave propagator in finite‐element modeling of tsunamis

Abstract

Finite‐element methods are often used to compute the evolution of tsunami waves over a grid suitably adapted to the selected basin. In this article it is shown that the finite‐element space discretization of the shallow‐water wave equations leads to a set of differential equations that are first order in time, and that, after linearization, admit a solution in terms of a linear operator we call the wave propagator. The initial perturbation is decomposed over the basis of the wave propagator eigenvectors, modified according to factors suitably depending on time and eventually synthesized as a superposition of the grid eigenmodes. The properties of the solution can therefore be analyzed in terms of the spectral components of the initial fields and of the spectral properties of the wave propagator. It is shown that a wave propagator can be defined even when the problem is constrained by means of the conditions imposed on the grid boundaries, which is indeed what occurs in all practical applications. Free and constrained wave propagator spectra are analyzed and discussed. The theory is applied to compute tsunami wave propagation in two simple cases: aflat‐bottom channel and a flat‐bottom, closed, rectangular basin. These examples are instructive since they permit us to introduce two simple methods for improving the numerical solution by controlling the numerical noise: annihilation of the exponentially growing spectral components and cutting of the high‐frequency eigenmodes.

Keywords:

• boundary conditions
• constraints
• finite elements
• spectral decomposition
• wave propagator