1
Chapter 1
Introduction
1.1 MATHEMATICAL MODELS
Hydraulic and coastal engineers are faced with complex practical and
theoretical problems that require extensive knowledge of hydrodynamics
and related environmental issues such as pollutant spreading and sediment
transport. In most cases, these problems can be adequately described and
solved by means of mathematical models.
Mathematical models are tools widely used to quantify cause–response
relations in a wide spectrum of applied disciplines, including engineering,
biology, economics and social sciences. These models can be deterministic,
stochastic or a combination of both, based on observational data and theoretical principles. The complexity and sophistication of the models can vary
from a simple statistical equation to a complex system of nonlinear partial
differential equations.
Mathematical models simulate the real world in a realistic but approximated manner. The approximate nature of all mathematical models is due
to the simplifying assumptions and parameterizations necessarily made to
reach a realistic mathematical formulation that accepts a feasible solution
(Shiflet and Shiflet 2014).
For simple models, the solution can be exact, like an analytical closedform solution, or approximate, like an open-form solution given as a series
with infinite number of terms. For complicated models, the solution can
only be obtained by numerical methods. Numerical analysis is the branch
of mathematics that deals with the development and evaluation of numerical methods. Mathematical models that are solved numerically, mostly by
engaging computers, are known as numerical models.
After the 1970s, extensive usage of numerical analysis and numerical
modelling in hydraulic engineering led to the development of computational hydraulics (Vreugdenhil 1981; Brebbia and Ferrante 1983; Hromadka,
Beech and Clements 1986; Abbott and Minns 1998). Computational hydraulics is part of the broader discipline known as computational fluid dynamics (CFD), which comprises of all branches of fluid mechanics but mainly
industrial flows and geophysical flows. Nowadays, applications of
Chapter 1
Introduction
1.1 MATHEMATICAL MODELS
Hydraulic and coastal engineers are faced with complex practical and
theoretical problems that require extensive knowledge of hydrodynamics
and related environmental issues such as pollutant spreading and sediment
transport. In most cases, these problems can be adequately described and
solved by means of mathematical models.
Mathematical models are tools widely used to quantify cause–response
relations in a wide spectrum of applied disciplines, including engineering,
biology, economics and social sciences. These models can be deterministic,
stochastic or a combination of both, based on observational data and theoretical principles. The complexity and sophistication of the models can vary
from a simple statistical equation to a complex system of nonlinear partial
differential equations.
Mathematical models simulate the real world in a realistic but approximated manner. The approximate nature of all mathematical models is due
to the simplifying assumptions and parameterizations necessarily made to
reach a realistic mathematical formulation that accepts a feasible solution
(Shiflet and Shiflet 2014).
For simple models, the solution can be exact, like an analytical closedform solution, or approximate, like an open-form solution given as a series
with infinite number of terms. For complicated models, the solution can
only be obtained by numerical methods. Numerical analysis is the branch
of mathematics that deals with the development and evaluation of numerical methods. Mathematical models that are solved numerically, mostly by
engaging computers, are known as numerical models.
After the 1970s, extensive usage of numerical analysis and numerical
modelling in hydraulic engineering led to the development of computational hydraulics (Vreugdenhil 1981; Brebbia and Ferrante 1983; Hromadka,
Beech and Clements 1986; Abbott and Minns 1998). Computational hydraulics is part of the broader discipline known as computational fluid dynamics (CFD), which comprises of all branches of fluid mechanics but mainly
industrial flows and geophysical flows. Nowadays, applications of