Propagation of tsunamis

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Table of contents

Introduction 
Propagation of tsunamis
Classical formulation
Dimensionless equations
Shallow-water equations
Boussinesq equations
Korteweg–de Vries equation
Energy of a tsunami
Tsunami run-up
1 Tsunami generation 
1.1 Waves generated by a moving bottom
1.1.1 Source model
1.1.2 Volterra’s theory of dislocations
1.1.3 Dislocations in elastic half-space
1.1.3.1 Finite rectangular source
1.1.3.2 Curvilinear fault
1.1.4 Solution in fluid domain
1.1.5 Free-surface elevation
1.1.6 Velocity field
1.1.6.1 Pressure on the bottom
1.1.7 Asymptotic analysis of integral solutions
1.1.8 Numerical results
1.2 Comparison of tsunami generation models
1.2.1 Physical problem description
1.2.2 Linear theory
1.2.3 Active generation
1.2.4 Passive generation
1.2.4.1 Numerical method for the linear problem
1.2.5 Nonlinear shallow water equations
1.2.6 Mathematical model
1.2.7 Numerical method
1.2.8 Numerical method for the full equations
1.2.9 Comparisons and discussion
1.2.10 Conclusions
1.3 Tsunami generation by dynamic displacement of sea bed
1.3.1 Introduction
1.3.2 Mathematical models
1.3.2.1 Dynamic fault model
1.3.2.2 Fluid layer model
1.3.3 Numerical methods
1.3.3.1 Discretization of the viscoelastodynamic equations
1.3.3.2 Finite-volume scheme
1.3.4 Validation of the numerical method
1.3.5 Results of the simulation
1.3.6 Conclusions
2 Dissipative Boussinesq equations 
2.1 Introduction
2.2 Derivation of the Boussinesq equations
2.2.1 Asymptotic expansion
2.3 Analysis of the linear dispersion relations
2.3.1 Linearized potential flow equations
2.3.2 Dissipative Boussinesq equations
2.3.3 Discussion
2.4 Alternative version of the Boussinesq equations
2.4.1 Derivation of the equations
2.5 Improvement of the linear dispersion relations
2.6 Regularization of Boussinesq equations
2.7 Bottom friction
2.8 Spectral Fourier method
2.8.1 Validation of the numerical method
2.9 Numerical results
2.9.1 Construction of the initial condition
2.9.2 Comparison between the dissipative models
2.10 Conclusions
3 Two phase flows 
3.1 Introduction
3.2 Mathematical model
3.2.1 Sound speed in the mixture
3.2.2 Equation of state
3.3 Formal limit in barotropic case
3.4 Finite volume scheme on unstructured meshes
3.4.1 Sign matrix computation
3.4.2 Second order scheme
3.4.2.1 Historical remark
3.4.3 TVD and MUSCL schemes
3.4.3.1 Green-Gauss gradient reconstruction
3.4.3.2 Least-squares gradient reconstruction method
3.4.3.3 Slope limiter
3.4.4 Diffusive fluxes computation
3.4.5 Solution interpolation to mesh nodes
3.4.6 Time stepping methods
3.4.7 Boundary conditions treatment
3.5 Numerical results
3.5.1 Convergence test
3.5.2 Falling water column
3.5.3 Water drop test case
3.6 Conclusions
4 Viscous potential flows 
4.1 Introduction
4.2 Anatomy of dissipation
4.3 Derivation
4.3.1 Dissipative KdV equation
4.4 Dispersion relation
4.4.1 Discussion
4.5 Attenuation of linear progressive waves
4.6 Numerical results
4.6.1 Approximate solitary wave solution
4.6.2 Discussion
4.7 Conclusion
Direction for future research
A VFFC scheme
A.1 Discretization in the finite volume framework
A.1.1 The one dimensional case
A.1.2 Extension to the multidimensional case
A.1.3 On the discretization of source terms
A.2 On the discretization of boundary conditions
Bibliography