Tsunami propagation for singular topographies

TL;DR

This paper develops a framework for analyzing tsunami waves over singular topographies, proving existence and uniqueness of solutions, and numerically exploring wave behaviors including a second wave phenomenon.

Contribution

It introduces a very weak solution concept for tsunami equations with singular coefficients and develops GPU algorithms for efficient computation.

Findings

Existence and uniqueness of very weak solutions are established.

Numerical experiments reveal a second wave traveling opposite to the singularity.

GPU algorithms significantly reduce computational costs for 2D simulations.

Abstract

We consider a tsunami wave equation with singular coefficients and prove that it has a very weak solution. Moreover, we show the uniqueness results and consistency theorem of the very weak solution with the classical one in some appropriate sense. Numerical experiments are done for the families of regularised problems in one- and two-dimensional cases. In particular, the appearance of a substantial second wave is observed, travelling in the opposite direction from the point/line of singularity. Its structure and strength are analysed numerically. In addition, for the two-dimensional tsunami wave equation, we develop GPU computing algorithms to reduce the computational cost.