2 Computational Modelling in Hydraulic and Coastal Engineering
computational hydraulics are ubiquitous. Proliferation of computational
hydraulics was supported by the fact that major hydraulic laboratories in
Europe and the United States switched from the cumbersome and expensive
physical hydraulic models to research, development and standardization of
mathematical-computational hydraulic models. Computational hydraulics
can effectively and efficiently solve and analyse problems pertaining to
water flow phenomena and design of hydraulic and coastal structures and
address environmental implications of pollutant transport by advection,
diffusion and dispersion (Chau 2010).
The numerical treatment of a mathematical model requires the synthesis
of a solution algorithm. The word algorithm describes a predetermined
sequence of basic arithmetic and logical operations for the solution of a
mathematical problem, where from a set of input data X, a set of output
(the solution in numerical form) data Y is produced.
Y = L(X)
(1.1)
where L is an operator. It is noticeable that most of the equations of hydraulic models belong to the category of linear homogeneous second-order
partial differential equations (PDEs). Consequently the subject of computational hydraulics can provide a unified view and understanding of the various fields of hydraulic engineering, including flows in closed conduits, open
channels and porous media (groundwater); waves and maritime hydraulics;
and pollutant and sediment transport (Abbott 1991).
1.1.1 Framework of numerical modelling
Theories on mathematical models and their numerical solutions are
quite extensive. An extremely synoptic presentation of their highlights is
attempted in the following (Fletcher 1991).
1.1.1.1 Well-posed mathematical models
A mathematical model of a system is considered to be well-posed if it has
the following:
• The solution algorithm produces a solution for all sets of input data
under specified conditions and limitations.
• The produced solution is unique, that is, only one solution output corresponds to each set of input data.
• The output has to be related to the input, via a Lipschitz condition,
that is, each infinitesimal change of input values (δX) results into a
finite change of output (δY).
computational hydraulics are ubiquitous. Proliferation of computational
hydraulics was supported by the fact that major hydraulic laboratories in
Europe and the United States switched from the cumbersome and expensive
physical hydraulic models to research, development and standardization of
mathematical-computational hydraulic models. Computational hydraulics
can effectively and efficiently solve and analyse problems pertaining to
water flow phenomena and design of hydraulic and coastal structures and
address environmental implications of pollutant transport by advection,
diffusion and dispersion (Chau 2010).
The numerical treatment of a mathematical model requires the synthesis
of a solution algorithm. The word algorithm describes a predetermined
sequence of basic arithmetic and logical operations for the solution of a
mathematical problem, where from a set of input data X, a set of output
(the solution in numerical form) data Y is produced.
Y = L(X)
(1.1)
where L is an operator. It is noticeable that most of the equations of hydraulic models belong to the category of linear homogeneous second-order
partial differential equations (PDEs). Consequently the subject of computational hydraulics can provide a unified view and understanding of the various fields of hydraulic engineering, including flows in closed conduits, open
channels and porous media (groundwater); waves and maritime hydraulics;
and pollutant and sediment transport (Abbott 1991).
1.1.1 Framework of numerical modelling
Theories on mathematical models and their numerical solutions are
quite extensive. An extremely synoptic presentation of their highlights is
attempted in the following (Fletcher 1991).
1.1.1.1 Well-posed mathematical models
A mathematical model of a system is considered to be well-posed if it has
the following:
• The solution algorithm produces a solution for all sets of input data
under specified conditions and limitations.
• The produced solution is unique, that is, only one solution output corresponds to each set of input data.
• The output has to be related to the input, via a Lipschitz condition,
that is, each infinitesimal change of input values (δX) results into a
finite change of output (δY).