Mathematical modeling for tsunami waves using lattice boltzmann method

Abstract

This research focuses on tsunami wave modelling. The nature of tsunami waves can be conditionally divided into three parts; generation, propagation and inundation (or run-up). General patterns and important characteristics of tsunamis can be predicted by various sets of governing equations and commonly used models which include elastic wave, nonlinear shallow water and forced Korteweg de Vries (fKdV) equations. In order to construct tsunami model, we divide this modelling into two parts; the first part contains seismic (earthquake) wave that focuses on the nonlinear elastic wave equation. The equation has been successfully applied to the tsunami generation part and is shown to give suitable complex flow simulation of elastic wave generation. The second part essentially deals with the nonlinear shallow water equations which are often used to model tsunami propagation and sometimes even the run-up part. This work specifically studies the properties of propagation of tsunamis. Shallow water equations have become the choice model of operational tsunami modelling for irrotational surface waves in the case of complex bottom elevation. The run-up part basically deals with the KdV and fKdV equations for unidirectional propagation and effects of external noise and damping terms for the studies of tsunami run-up. Several test-cases are presented to verify propagation and run-up model. The simulation algorithm of this research is based on the lattice Boltzmann method (LBM). The aim of this research is to use the LBM to solve tsunami waves modelling. Several problems for simulation of tsunami waves are generated with LBM. The appropriate equilibrium distribution function is selected and extended to solve the related three-dimensional problems and appropriate units are chosen and changed in accordance with lattice Boltzmann simulations and stability of lattice Boltzmann models. These models are solved and the solutions with different boundary conditions are analysed to produce relevant patterns and behaviours, assumptions and approximations for modelling tsunami and seismic waves. These analyses have been implemented via accurate, robust and efficient LBM for solving the tsunami sets of equations under complex geometry and irregular topography. The graphical output profiles are generated by using Matlab version 2012.