Complete classification for the (1+1)D non-linear boussinesq equations
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Date
2020
Authors
Carvalho, Cindy
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Abstract
This thesis contributes to the knownanalytical solutions of the (1+1)D non-linearBoussinesq
(NB) equations, which effectively describe solitary wave motion in shallow water
(typically used to describe tsunami wave motion) using a systematic symmetry approach.
These results can be used to enrich the analytical validation ofnumerical models for tsunami
detection systems.
Tsunami detection systems based on numerical models provide real-time forecasting
and inundation mapping, playing a critical role in evacuation plans. These models have
become more sophisticated over the past three decades through precise and rigorous analytical
and experimental validation. Although analytical and experimental benchmarking
cannot ensure that numerical models that pass benchmark tests will always produce realistic
inundation predictions, validation largely reduces the uncertainty in the results to
initial geophysical conditions [1].
The true benefit of analytical solutions in tsunami model predictions is the ability to
identify the dependence of the desired results (such as run-up height, wave height and
wave speed) on the independent variables of the model (such as water depth and beach
slope). Problem scaling is also made simpler through analytical calculations, as opposed to
numerical techniques, which usually require repeated numerical computations. Lastly, by
comparing the numerical solutions to exact analytical solutions, systematic errors can be
identified, making analytical solutions useful in validating complex numerical models with
realistic applications [1]. The use of conserved quantities as an analytical validation mechanism
is a good example of how analytical solutions can assist with numerical tsunami
model validation. Upon substitution of these quantities into the numerical models, these
conserved quantities should remain constant throughout the calculation. Although analytical
solutions are powerful for the reasons previously stated, these solutions are complex
to obtain, and therefore, not as widely used in comparison to experimental validation for
tsunami detection system development.
The (1+1)DBoussinesq equations investigated in this work are considered for the equa-
tions’ ability to model tsunamis via solitary wave motion. Three models are investigated,
namely the (1+1)DBoussinesq equations for amoving bathymetry, the (1+1)D linearBoussinesq
(LB) equations at a constant water depth and the NB equations at a constant water
depth.
We obtain three cases of solutions for the NB equations at a constant water depth dependent
on the wave speed c 2, namely (i) 0 < c 2 < 1, c 2 6= 1
15 and c 2 6= 1
3 ; (ii) c 2 = 1
15
and (iii) c 2 = 1
3 . To our knowledge, the invariant exact travelling wave solutions for the NB
equations for cases (i) and (iii) are new and can be used in analytical validation of numerical
models for tsunami detection systems, whilst the remaining case, case (ii), is identical
to that displayed in the work of Dutykh and Dias [2].
We provide physically realistic explanations of the results relating to the horizontal
component of the fluid velocity and surface elevation of the solitary wave for the invariant
exact solutions. This is done by considering seven cases dependent on the wave speed
c 2, namely 0 < c 2 < 1
15 , c 2 = 1
15 , 1
15 < c 2 < 1
3 , c 2 = 1
3 , 1
3 < c 2 < 1, c 2 = 1 and 1 < c 2 <1. To our
knowledge, these are newly discovered solutions and results.
Description
A thesis submitted in fulfillment of the requirements for the
degree of
Doctor of Philosophy
in the Faculty of Science at the University of the
Witwatersrand, Johannesburg, 2020