Environmental Engineering Reference
In-Depth Information
may be described by a 1D approach, if relevant conditions do not change substan-
tially in the vertical direction and along the shoreline.
1D steady state models lead to ordinary differential equations. Transient models,
including at least one space direction, lead to partial differential equations. It is
important to know about these differences, as mathematical solution techniques for
both types of equations are different and different MATLAB
®
commands need to
be used. Here we use MATLAB
for steady and unsteady modeling in 1D.
2D models include two space variables. One may distinguish between 2D
horizontal and 2D vertical models. Terrestrial ecology is a typical field, where
this type of model is suitable, describing the distribution or population of species on
the land surface. In streams or estuaries or in shallow water models are often set up
for vertically averaged variables, for which a 2D horizontal description results.
Models for 2D vertical cross-sections are obtained,
®
In groundwater flow, where several geological formations are to be included, but
no variations of hydraulic conditions in one horizontal direction
In cross-sections of streams
In air pollution modeling, if no space direction is preferential around a source; in
that case the single radial coordinate
r
replaces two horizontal space variables
x
and
y
3D models are quite complex in most cases and will marginally appear in this
book, as the focus is on simple explanatory examples. Numerical algorithms using
the methods of Finite Differences, Finite Volumes or Finite Elements are the
methods of choice for modeling in higher space dimensions, steady and unsteady.
The MATLAB@ 'Partial Differential Toolbox' can be recommended for the
application of these algorithms.
1
As the focus here is on core MATLAB
®
,we
leave numerical methods out. Instead it is outlined, how core MATLAB
can be
applied for steady state modeling in higher dimensions, based on computing of
analytical solutions.
®
2.2 Modeling Steps
The task of modeling can be sub-divided in several steps. The way from a real
system to the working model contains different tasks, where every step depends on
good performance of the previous step. The major steps are to build a conceptual
model, to describe it by mathematical analysis, to solve the differential equations by
computational methods and finally to perform post-processing tasks. A schematic
overview of the procedure is given in Fig.
2.1
(see also: Holzbecher
1998
).
1
In the partial differential equations toolbox, the modeling of advection processes (see below) is
difficult and requires numerical skills. All other processes can be simulated using the appropriate
commands.
