3 Modelling concepts
Everywhere in this chapter we will use a term “modelling” in the sense of “numerical
modelling”. Physical modelling and conceptual modelling as well as numerical modelling itself
will be mentioned directly in relevant cases.
3.1
Introduction
In chapter 3 the concept of transport equation was introduced, starting from the
concepts of control volume and of accumulation rate of a property inside that control
volume. Diffusive and advective fluxes were also defined to account for exchanges
between the control volume and neighbourhood and the concept of evolution equation
was introduced adding sources and sinks to the transport equation. A model is built on
the same concepts. Its implementation needs the definition of at least a control
volume, the calculation of the fluxes across its boundary and the calculation of the
source and sinks using values of the state variables inside the volume.
The number of dimensions of the model depends on the importance of relevant
property gradients. The simpler model is the "zero-dimensional" model. In this case,
there is no spatial variability and only one control volume needs to be considered. On
the other extreme of complexity is the three-dimensional (3D) model, required when
properties vary along the 3 space dimensions.
Whatever is the number of dimensions of a model, it must include the following
elements:

Equations,

Numerical algorithm,

Computer code.
The order of the items in this list can also be seen as their chronological development
order. Hydrodynamic equations are based on mass, momentum and energy
conservation principles, which were presented in chapter 3 and are known for longer
than one century. Numerical algorithms to solve hydrodynamic models have been
attempted even before the existence of computers. The analytical equations and the
numerical algorithms produced before the existence of computers allowed the fast
development of modelling since the sixties, when computers were made available to a
small scientific community. Since that time models and the modelling community has
evolved exponentially. Modern integrated computer codes have done more for
interdisciplinarity than one century of pure field and laboratory work.
After a period of validation of model's assumptions (equations and algorithms) the
development of user-friendly graphical interfaces becomes a priority, easing the use
of models by non-specialists. They reduce the number of errors on input files, and
make easy the checking of those files. Along this chapter concepts and methodologies
to build models and to understanding their functioning are being presented.
3.2
Numerical discretisation techniques
Computers can only solve algebraic equations. Analytic equations - integral or
differential - must be discretised into an algebraic form. The procedure to follow
depends on the form of the analytical equation to be solved. The control volume
approach is the most adequate to the integral form of evolution equations, while
Taylor series are the most adequate for differential equations.
3.2.1 Computation grid
The calculation of fluxes across control volume surface is also simpler if the scalar
product of the velocity by the normal to each elementary area (face) composing that
surface remains constant along the domain. The control volume that makes that
calculation simpler must have faces perpendicular to the reference axis. If rectangular
coordinates are used, the control volume generating the simpler discretisation is a
parallelepiped. In case large oceanic/atmospheric models a suitable control volume
must have faces laying on meridians and parallels (geographic coordinates).
In depth-integrated models the upper face of the control volume is the free surface and
the lower face is the bottom. In 3D models a control volume occupies only part of the
water column and its shape depends on the vertical coordinate used. In coastal
lagoons Cartesian and sigma type coordinates (or a combination of both) are the most
common.
The ensemble of control volumes forms the computation grid. In finite-difference type
grids control volumes are organized along spatial axis and a structured grid is
obtained. On the contrary, typical finite-element grids are non-structured. The latter
are more difficult to define, but more flexible, allowing easy variability of spatial
resolution. Figure 3.5-1 shows an example of a very general finite-difference type grid
using several discretisations on vertical direction.
Figure 3.5-1: Example of a grid for a 3D computation. Two
vertical domains are used. In the upper domain, a
sigma coordinate is used while lower one uses a
Cartesian.
Figure 3.5-2: Typical 1D spatial grid.
3.2.2 The 1D model case
Figure 3.5-3: Generic control volume in a 1D discretisation.
A system can be considered as one-dimensional if properties do change along a
physical dimension mostly. In this case control volumes can be aligned along the line
of variation and one spatial coordinate is enough to describe their locations. Properties
are considered as being constant across control volume faces perpendicular to that
axis. Fluxes across faces not perpendicular to that axis are null or negligeable.
Vi
Vi-1
Vi+1
Figure 3.5-1: Example of a grid for a 3D computation. Two vertical
domains are used. In the upper domain, a sigma coordinate
is used while
lower one uses a Cartesian.
3.2.2.1 Control volume
approach
Control
volumes
Figure
3.5-2:
derivation of the
Figure 3.5-3:
generates simple
used by1D
numerical
models have the same meaning as in the
Typical
spatial grid.
evolution equation in chapter 3. A discretisation is adequate if it
Generic control volume in a 1D discretisation.
calculation algorithm, still keeping results accuracy. The simpler
calculation is obtained if properties can be considered as being uniform inside the
control volume and along parts of its surface (each face). To make this possible
Q i-½
Ci-1
Ci
 i-½
Q i+½
 i-½
Vi
Vi-1
Vi+1
Δxi
Δxi-1
Ai-½
Figure 3.5-3: Generic
Ci+1
Δxi+1
Ai+½
control
volume
in
a
1D
discretisation.
without compromising accuracy, the control volume must be as small as possible: a
fine grid is needed.
In a 1D model properties can be stored into 1D arrays (vectors). Neighbours of a
generic element "i" is "i-1", on the left side and "i+1" on the right (Figure 3.5-3). The
length of a control volume must be small enough to allow properties in its interior to
be represented by the value in its centre. In that case equations deduced in paragraph
3.2 apply and the rate of accumulation in volume “i” will be given by:
AccumulationRate 
Vi Ci t  t  Vi Ci t
t
Δt is the time step of the model. This equation simplifies if the volume remains
constant in time. This is not the case in most coastal systems and is not certainly the
case in tidal systems.
Exchanges between "i" volume and neighbours are accounted by advective and
diffusive fluxes. Their calculation needs some hypotheses. Let us detail Figure 3.5-2,
indicating the distances between faces (spatial step) and the location points where
others auxiliary variables are defined. Advection fluxes at volume “i” contribute with:
Q C
i
i  12
 Qi 1Ci  12

t t *
where Qi=uiAi-½ and diffusive flux contributes with:
 
  i  12
 C  Ci 1 

Ai  12  i
1
 2 xi  xi 1  
t t *
 
  i  12
 C  Ci 

Ai  12  i 1
1
 2 xi  xi 1  
t t *
In these equation t* is a time instant between t and t+t, to be defined according to
criteria defined in the next paragraph.
Adding the 3 contributions described above one gets:
Vi Ci t  t  Vi Ci t
t

 Qi Ci  12  Qi 1Ci  12
 
 Ci  Ci 1 

Ai  12 
1 x  x

2
i
i 1 

i  12
t t *

t t *

 
  i  12
 Ci 1  Ci 

Ai  12 
1 x  x

2
i
i 1 

t t *
(Equation 3.5-1)
In the particular case of a channel with uniform and permanent geometry (cross
section (A), volume (V) constant) and with constant discharge and diffusivity
Equation 3.5-1 becomes:
Ci
t
 Ci  1 2  Ci  1 2
 Ci
 U
t
x

t  t




t t *
 C  2Ci  Ci 1 
   i 1

x 2


t t *
Where U is the cross-section average velocity and x is the ration between the
volume and the average cross section (i.e. is the length of the control volume). This is
a most popular form of the transport equation, but as shown above, it is applicable
only into particular conditions.
Additional approaches are required to calculate the advective flux, since concentration
is defined at the centre of the control volumes and not at the faces. Those approaches
ant their numerical consequences are described in the next sections.
3.2.2.2 Numerical calculation of advection
Three common approaches to estimate concentration values at control volume faces
are:

Linear Approach,

Upstream stepwise approach,

Quadratic Upwind Approach (QUICK).
Linear Approach
In the linear approach it is assumed that:
Ci  1 
2
Ci xi 1  Ci 1xi
xi 1  xi
Assuming a discretisation where the grid size is uniform, it is easily seen that this
approach generates central differences as obtained using Taylor series (see next
paragraph).
Upstream stepwise approach
In this case it is assumed that the concentration at left face is:

Q  0  C
Qi  0  Ci  1  Ci 1
i
2
i  12
 Ci


This discretisation respects the transportivity property of advection. This property
states that advection can transport properties only downstream and never upstream.
The linear approach doesn’t respect this property because volume “i” will get
information of downstream concentration through the average process. The violation
of this property can generate instabilities and will create conditions to obtain negative
values of the concentration. The upstream discretisation avoids that limitation but as
shown into next paragraphs can introduce unrealistic numerical diffusion.
Quadratic Upwind Approach (QUICK)
The quadratic upwind scheme aims to compromise the respect of the transportivity of
advection and numerical diffusion (explained further down). In this case it is assumed
that concentration distribution around a point follows a quadratic distribution centred
on the upstream side of the face being calculated. For the left face one would get:

Q  0  C
Qi  0  Ci  1  6 8 Ci 1  3 8 Ci  18 Ci 2
i
2
i  12
 6 8 Ci 1  3 8 Ci  18 Ci 1


Using the Taylor series discretisation described in the next paragraph, it can be seen
that advection calculated using this approach is 3rd order accurate (Leonard, 1976),
while pure upstream discretisation is 1st order accurate and the linear approach
(central differences) is 2nd order accurate. The inconvenience of QUICK discretisation
is that it requires additional approaches close to the boundaries. This is not a very
much limiting factor in 1D calculation, but it is in 2D or 3D calculations, especially
when geometry is irregular.
3.2.2.2.1 Temporal approach
In previous paragraphs spatial discretisation was analysed. A solution was described
for diffusion term and three discretisation methods were suggested for advection, but
Property value
140
C
0
1
t
Time
Figure 3.5-4: Visualization of the consequences of temporal
discretisation. Property evolves within a time
step, but not values used for flux calculation.
nothing was said about the instant of time of the variables used to calculate advection
or diffusion. Figure 3.5-4 shows an example of a time evolution of a property in a
space point. The curve line shows the continuous evolution and dots show values at
each time step. Figure shows values at the beginning and end of a particular time step.
The flux in that time step is proportional to the dark area shown on the figure. Values
at the beginning and end of a time step are shown, as well as concentration variation
during that time step. The rate of accumulation at this point is proportional to the
slope of this line. The slope of this line also gives an idea of the errors associated to
the choice of t*. Explicit models use t*=t, while implicit models consider t*=t+t.
From that figure it can be seen that when the slope of the curve is positive explicit
models underestimate the advective fluxes1, while when the slope is negative they
overestimate them, introducing a phase error. On the contrary, implicit methods
overestimated the fluxes by a value of the same order and also introduce a phase error
(but with opposite sign). The consideration of an intermediate value between t and
1
In explicit methods the flux during a time step is proportional to the are of the rectangle with sides
lengths t an Ct, while in implicit methods is proportional to t an Ct+t.
t+∆t generates more accurate fluxes. In the next paragraph it will be shown that t*=
t+½ t (semi-implicit method) gives the maximum accuracy. Values at t*= t+½ t
can be obtained averaging the values of the properties calculated at time t and time
t+t. The price to pay for this accuracy improvement is the increasing of the number
of calculations to perform.
In next paragraph it will be shown that implicit methods have better stability
properties than explicit methods, and it can be shown that semi-implicit method’s
stability properties are similar to those of implicit methods. For their stability and
accuracy properties, semi-implicit methods are the most efficient numerical methods.
3.2.2.3 Taylor series approach
Traditionally discretised equations are obtained from partial differential equations
replacing derivatives by finite-differences obtained using Taylor series. Taylor series
provide information on the truncation errors done when replacing derivatives by
finite-differences. On the contrary the control volume introduced in previous
paragraph gives information about physical approaches done during discretisation.
Done correctly both methodologies must produce the same discretised equations.
To introduce the Taylor series discretisation methods and to analyse stability and
accuracy concepts, let us consider the differential equation corresponding to Equation
3.5-1:
C
C
 2C
U
 2
t
x
x
(Equation 3.5-2)
This equation describes advection-diffusion transport in a channel with uniform
velocity, permanent geometry and diffusivity.
3.2.2.3.1 Time discretisation (g)
Taylor series relate the value of a property in a point (or time instant) with the values
of the property in another point and the derivatives in the same point:
C
t
t
t
2
2
3
3
 t n  n C 
 C   t  C   t  C 




  0t n 1
 C   t




  
2 
3 
n 

 t  i  2 t  i  3! t  i
 n! t  i
t
t  t
i
t
i
Truncating this series at the first derivative, one gets:
t  t
 C  Ci  Ci
 t 

 
t
 t  i
t
t
Equation 10
This equation states that the resolution all the terms of the equation at time t allows
the calculation of the variable at time t  t with precision of first order, since the first
missing term in the series is multiplied by t .
Similarly one could relate the concentration at time t with the concentration at
time t  t :
t  t
t  t
t  t
t  t
 t 2  2 C 
 t 3  3C 
 t n  n C 
 C 





C C
  t


   

2 
3 
n 
2
3
!
n
!

t

t

t
 t  i

i

i

i
Truncating this series after the first derivative as before, one gets:
t  t
i
t
i
t  t
 C 


 t  i

Cit  t  Cit
 0t 
t
 0t 
Equation 11
This equation shows that in implicit methods the truncation error is also of the first
order, as in explicit methods, although processes are computed at time t  t . Below it
will be shown that difference between implicit and explicit methods are their stability
properties.
From paragraph 3.2.2.2.1 it was expected that explicit and implicit should have the
same truncation error and it is also expected that the calculation of the derivatives (or
fluxes) at the centre of time step must have a smaller truncation error. To demonstrate
it, let us use Taylor series to relate properties at time t and t  t with variables at
t  t / 2 .
t  t / 2
C 

C it  t  C it  t / 2   t

2
t  i

t  t / 2
C C
t
i
t  t / 2
i
C 

  t

 2 t  i
 
 t

 2
 2

 
 t

 2
 2

2
2
t  t / 2

 C

t 2 
i
2
 2
 0 t
3
Equation 12
t  t / 2

 2C 

t 2 
i
 2
 0 t
3
Subtracting the second equation from the first equation, one gets:
t  t / 2
 C 


 t  i
 
Cit  t  Cit

 0 t
2
t
2
Equation 13
n 1
This equation shows that semi-implict methods are second order accurate, and
consequently they allow for the use of larger values of time step. The implementation
of these methods requires the computation of all derivatives and fluxes centred in
time. The values to be used in the calculations can be computed also with second
order accuracy, as the average between values at time t and t  t , as can be
demonstrated using expansions from Equation 12:
Cit  t / 2 
Cit  Cit  t
2
 0t 
2
This temporal semi-implicit discretization is known as a the Crank-Nicholson
discretization. In this discretization one would get:
t  t
Cit  t  Cit 1 
C
 2C 
   U
  2 
t
2
x
x  i
t
1
C
 2C 
   U
  2   0(t ) 2
2
x
x  i
To solve this equation, spatial derivatives must be discretised.
3.2.2.3.2 Spatial discretisation
Spatial discretization using Taylor follows an approach similar to temporal
discretisation. Let’s consider Taylor series developments for points on the left and on
the right of point i, at a distance x :
*
*
C
*
i 1
2
2
3
3
 C   x  C   x  C 
  
  0x 3
 C   x
  
2 
3 
x  i  2 x  i  3! x  i

C
*
i 1
2
2
3
3
 C   x  C   x  C 


  0x 3
 C   x

  
2 
3 

x  i  2 x  i  3! x  i

*
t
i
*
*
t
i
Equation 14
*
Equation 15
Subtracting Equation 15 from Equation 14, one gets the so called central difference
for first order spatial derivative of C:
Cit1  Ci*1
 C 
2
 0x 

 
2x
 x  i
*
Equation 16
From Equation 14, one would get an expression for a non-centred derivative (right
side derivative), and from Equation 15 a left side derivative, both with a truncation
error of first order:
Cit1  Ci*
 C 

 0x 


x
 x  i
Equation 17
C t  Ci*1
 C 
 0x 

  i
x
 x  i
Equation 18
*
*
If Equation 17 is used when the velocity is negative and Equation 18 is used when the
velocity is positive, the first derivative is computed using an “upstream method”,
since in both case no downstream information is used.
Adding Equation 14 and Equation 15, one gets:
*
  2C 
C *  2Cit  Ci*1
2
 2   i 1
 0x 
2
x
 x  i
Equation 19
Which is the finite-difference form of the second spatial derivative, discretised with a
second order truncation order.
In next paragraph stability criteria is analysed for some of these discretizations. It will
be shown that central differences for first order derivative generate unstable
algorithms and it will be shown that truncation error is not the unique aspect to take
into account for estimating the accuracy of a numerical algorithm.
3.2.2.4 Stability and accuracy
For simplicity let us consider Equation 3.5-2 to illustrate stability and accuracy
associated to different options for temporal and spatial discretisation. Let us consider
central explicit differences in the particular case of no diffusion. In that case Equation
3.5-2 becomes:
t  t
 Ci
 C  C i 1 
 U i 1

t
x


Ut
Ut
t  t
t
t
t
Ci
1
C  Ci  1
C
2 x i 1
2 x i 1
Ci
Where
t
t
(Equation 3.5-6)
Ut
 C r is the Courant number, representing the ratio between the path
x
length of a particle during a time step and the grid size. This is a critical parameter for
explicit discretisation methods.
Let us consider the case of a channel where initial concentrations are zero everywhere
except in a generic point “i”. Table 3.5-1 shows the temporal evolution along 11 time
steps (0 to 11) for the case of unitary Courant number and central explicit differences
Table 3.5-1: Example of a time evolution in a 1D channel computed using explicit
central differences, a unitary courant number, and no diffusion.
Time step i-3
0
1
2
3
4
5
6
7
8
9
10
11
i-2
0
0
0
0
0
0
0
0
0
0
0
0
i-1
0
0.00
0.25
0.75
1.31
1.56
1.08
-0.33
-2.29
-3.76
-3.26
0.31
Grid point number
i
i+1
0
1
-0.50
1.00
-1.00
0.50
-1.13
-0.50
-0.50
-1.63
0.97
-2.13
2.81
-1.16
3.93
1.66
2.94
5.59
-1.00
8.52
-7.14
7.52
-12.54
0.38
i+2
0
0.50
1.00
1.13
0.50
-0.97
-2.81
-3.93
-2.94
1.00
7.14
12.54
Total
amount
i+3
0
0.00
0.25
0.75
1.31
1.56
1.08
-0.33
-2.29
-3.76
-3.26
0.31
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
for advection and Table 3.5-2 shows the corresponding solution for the case of a
Cr=2.
In both tables column “i-3” and “i+3” represent the boundary conditions (zero outside
of the modelling area) and total amount stands for the total amount of matter inside
the channel. Both solutions are unrealistic.
In such conditions one would expect the contaminated water to move forward and
after a certain time whole the channel should have a concentration equal to zero
because the water entering in the model area has null concentration. The value of the
total amount of matter inside the channel should remain constant.
Stability
A model is said unstable if errors generated inside the modelling area are amplified.
This is what happened in both calculations. As time evolved the errors have grown.
Error growth rate has been higher at higher Courant number.
To understand the reasons for instability one can use the following principle:
“The influence of a point on its neighbours through advection or diffusion can’t be
negative”.
This means that the consequence of increasing the concentration in one point can
never be a reduction in any of its neighbours. Increasing the concentration in none
point will result on an increase in the neighbours ot it will have no effect on them n(if
advection and/or diffusion
have no capacity to transport material to there. To
guarantee the respect of this principle, no coefficient multiplying grid point values in
Equation 3.5-6 can be negative. If a coefficient is null there is no influence from that
point on point “i”. In Equation 3.5-6 the coefficient of Ci+1 is negative whatever is the
Courant number. As a consequence the higher is the concentration in that point,
smaller becomes the concentration in point “i”.
Adding diffusion this method can be stabilised. Considering diffusion, Equation 3.5-6
would become:
t  t
t
t
 Ci
 C  C i 1 
 C  2C i  C i 1 
 U i 1
    i 1

t
x
x 2




Equation 3.5-7
t  t  1 Ut t  t
 1 Ut t 

t  t
t
Ci

 2 C i 1  1  2 2 C i   

C
2 x x 2  i 1
x 
 2 x x 


Ci
In
t
t
x 2
 d is called the diffusion number. In this case positiveness of the
Table 3.5-2: Example of a time evolution in a 1D channel computed using explicit
central differences, Cr=2, and no diffusion.
Time step i-3
0
1
2
3
4
5
6
7
8
9
10
11
i-2
0
0
0
0
0
0
0
0
0
0
0
0
Grid point number
i
i+1
i+2
i+3
0
1
0
0
-1.00
1.00
1.00
0.00
-2.00
-1.00
2.00
1.00
0.00
-5.00
0.00
3.00
8.00
-5.00
-8.00
3.00
16.00
11.00
-16.00
-5.00
0.00
43.00
0.00
-21.00
-64.00
43.00
64.00
-21.00
-128.00
-85.00
128.00
43.00
0.00
-341.00
0.00
171.00
512.00
-341.00
-512.00
171.00
1024.00
683.00 -1024.00
-341.00
Total
amount
i-1
0
0.00
1.00
3.00
3.00
-5.00
-21.00
-21.00
43.00
171.00
171.00
-341.00
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
coefficients is assured if:
Re g 
Ux

2
Ut
Cr 
1
x
(Equation 3.5-8)
Reg use to be designated by “Grid Reynolds number” The consideration of advection
alone is equivalent to the consideration of an infinity Reynolds number and
consequently, whatever is the time step (or Cr), central-differences are always
instable.
The consideration of diffusion doesn’t always increase the stability properties of
numerical models. Why did it in this case? Central differences do not respect the
transportive property of advection. Physically, advection can only propagate
information on the sense of the velocity. The analysis of Table 3.5-1 and Table 3.5-2
shows that information has been also propagated backward. This was a consequence
of the use of a downstream value (Ci+1) to calculate the spatial derivative. Physically
diffusion propagates the information in any direction (according to the local
gradients). In case of Table 3.5-1 and Table 3.5-2 diffusion transports matter
upstream, making it available to be transported subsequently by advection and thus it
increases stability.
When the advective flux is calculated using downstream information, one can remove
matter from a control volume that is not there to be removed. This is the mechanism
that generates negative concentrations. The method is unstable because those errors
are amplified in time. The consideration of (enough) diffusion makes the method
stable, but do not avoid the generation of negative concentrations. The upstream
discretisation was proposed first to avoid this problem.
Let us consider now upstream explicit differences and again the particular case of no
diffusion. In that case Equation 3.5-2 becomes:
Ci
t  t

Ut
Ut  t

t
Ci 1  1. 
Ci ( forU  0)
x
x 

(Equation 3.5-6)
In this case it is easy to verify that the method is stable if the Courant number is not
greater than 1. Table 3.5-3shows results for Cr=1 and Table 3.5-4 for Cr=0.5. Cr>1
Table 3.5-3: Example of a time evolution in a 1D channel computed using explicit
upstream differences, Cr=1.0, and no diffusion.
Time step i-3
0
1
2
3
4
i-2
0
0
0
0
0
i-1
0
0.00
0.00
0.00
0.00
Grid point number
i
i+1
0
1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
i+2
0
1.00
0.00
0.00
0.00
Total
amount
i+3
0
0.00
1.00
0.00
0.00
0
0
0
0
0
1
1
1
0
0
Table 3.5-4: Example of a time evolution in a 1D channel computed using explicit
upstream differences, Cr=0.5, and no diffusion.
Time step i-3
0
1
2
3
4
5
6
7
8
9
10
i-2
0
0
0
0
0
0
0
0
0
0
0
i-1
0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Grid point number
i
i+1
0
1
0.00
0.50
0.00
0.25
0.00
0.13
0.00
0.06
0.00
0.03
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
i+2
0
0.50
0.50
0.38
0.25
0.16
0.09
0.05
0.03
0.02
0.01
i+3
0
0.00
0.25
0.38
0.38
0.31
0.23
0.16
0.11
0.07
0.04
0
0
0
0
0
0
0
0
0
0
0
Total
amount
1.00
1.00
1.00
0.88
0.69
0.50
0.34
0.23
0.14
0.09
0.05
would generate an instable model, which could not be solved adding diffusion. In fact
if diffusion was considered, the stability criteria would be (Cr+2d)1.
In Table 3.5-3 one can that explicit upstream differeces give the exact result. The
concentration reamins constant and travel at the exact speed (1 cell per iteration).
When the Courant number is reduced to 0.5 (Table 3.5-4) the solution is however
degradated through the introduction of numerical diffusion. The method remains
stable since the errors are reduced in time.
The results obtained in the 4 examples presented show that small truncation errors as
given by the Taylor series are not enough to guarantee accurate results. The upstream
results show that the reduction of the time step do not guarantee either an
improvement of the results.
The need for a fine resolution grid
The reason why the upstream scheme with a Cr=0.5 gives so poor results is the coarse
discretisation used. In this case the matter travels only half of the grid size and
consequently the material contained in cell “i” at t=0 is dritibuted by two computing
cells at time t=1. Because the concentration is computed as the mass devided by the
volume, its values is reduced to ½. This resulot is obtained because we are violating
an initial hypothesis “the grid cell is small enough to allow the concentration to be
uniform in its interior”. This is not happening when half of the cell has matter and not
the other half. If the plume was contained into many cells the problem would still
exist but only on the plume limits and would not deteriorate the solution.
3.3
Model implementation (r+f)
As described in chapter 3, the implementation of a numerical model in a specific
lagoon requires the specification of the boundary conditions specific of the site and
the specification of model parameters suitable for the site and for the conditions
envisaged for the simulations.
The bathymetry and the tidal harmonics at the inlet(s) are boundary conditions that
can be considered as time independent in most sites. On the contrary river discharge,
water column structure at the sea boundary (in case of 3D simulations) and the
atmospheric conditions, are time dependent boundary conditions.
Implementation of a model requires data to specify boundary conditions, but also data
to specify initial conditions and parameters for process simulation. Additional data is
required to evaluate of model results quality.
3.3.1 Bathymetry and modelling grid
Bathymetry describes lagoon’s geometry and is the basis of the whole modelling
procedure. Bathymetry is generally is measured by Hydrographic Authorities using a
fine resolution to provide data for navigation charts and to support coastal engineering
works. In terms of bathymetry and grid definition laterally integrated models
constitute a particular case, requiring much less information.
3.3.1.1 1D models
In the beginning of this chapter it was shown how to build a 1D model and the
parameters characterizing the respective grid: cross section and cell length. In that
case the volume of each cell was calculated as the product of the cross section by the
cell length. As a general rule the free surface level varies in time and the model has to
calculate the value of the cross section as a function of that level. In these conditions,
the information to supply is the width of the cross section as a function of the level.
This information has to be supplied at both tops of each cell or at its centre.
Intermediate values can be calculated interpolating.
3.3.1.2 Horizontally resolving models
Horizontal resolution models are 2D depth integrated or 3D resolving models. In
these models the modelling area is described by a grid usually formed by rectangular
cells in finite-difference methods and by triangles in finite-element methods. The
depth is specified in the centre of the grid or at the corners and obtained by
interpolation in other points. In this type of models horizontal resolution is much
higher that in laterally integrated models and more detailed bathymetric information is
required. In general “grid-generating programs” create the grids. Those programs
require the supply of a detailed coastline and obtain the depth of each cell
averaging/interpolating a digitised bathymetry. When the information is scarce,
special care has to be taken verifying the depth generated for each model cell.
In case of 3D models, after the generation of the horizontal grid and of the
corresponding depth distribution, a vertical discretisation has to be defined. Two
discretisations are easily defined: Sigma-type coordinates and Cartesian coordinates.
In Cartesian coordinates layers are horizontal and maintain the same thickness in
time, apart from the surface layer, which thickness depends on the free surface
position. In case of sigma coordinates the water column is divided into layers of
variable thickness (in space and time) in such a way that the ratio between the
thickness of a layer and the local depth remains constant in space and time. In sigma
type models vertical resolution is independent of the local depth. This is convenient in
systems where density effects play a secondary role compared with topographic
effects. Special care has however to be taken in intertidal areas, where the thickness of
each layer approaches zero during the drying procedure. The consideration of two
sigma domains in the water column can be a solution.
3.3.2 Initial conditions
Initial conditions must be supplied for each state variable (velocity and levels,
temperature, salinity, nutrients, etc.). This information should be obtained from field
data measured synoptically. Unfortunately for most variables information is unknown
or available in just a few points and assumptions must be done. For that purpose
properties have to be grouped into rapidly dissipative and slowly (or non) dissipative
properties.
Dissipative properties rapidly forget the information related to the initial conditions
becoming dependent only on the boundary conditions. Hydrodynamics is driven by
mechanical energy and is highly dissipative. In fact errors on the hydrodynamical
initial conditions result into artificial initial energy supply, which will be transformed
into kinetic energy before being dissipated. Stability properties of the model are
limited by the values of the velocity. Very unrealistic initial conditions can generate
very high velocities and create conditions for numerical instability. For these reasons
the most efficient way to initialise a hydrodynamic model is to start from null velocity
and a horizontal free surface equal to the average level at the open boundaries.
Properties associated to the ecology of the system have to be initialised with realistic
values. In fact ecological systems are resilient and after a strong perturbation do not
necessarily recover to the state before that perturbation. Initialising and ecological
model with unrealistic values is equivalent to state that the system has been highly
disturbed. In fine resolution models a lot of information must be supplied
interpolating from point information available. Specific initialisation software tools
can simplify the initialisation procedure.
Sediment transport models are also very much independent on the initial conditions. If
initial suspended matter concentration is specified in excess, settling will remove it.
On the contrary if initial suspended matter is underestimated erosion will supply the
missing matter (if deposited matter is available). In this case the difficulty is not the
initialisation of the water column concentration, but the initialisation of the erodible
amount of sediments lying on the bottom. The user must use a procedure compatible
with the information available and with the objectives of the simulation.
If the information is scarce and the aim of the simulation is to obtain realistic values
of the concentration in the water column, then a convenient procedure is to assume
that there is no net erosion in any point of the lagoon. In this case the model is
initialised assuming that there is a thin erodible layer everywhere in the lagoon and is
run until an equilibrium solution is obtained. After this period erosion and deposition
areas are identified and the model is initialised assuming that there is no erodible
sediment in the areas where erosion was identified.
If the aim of the model is to simulate erosion processes in the lagoon (e.g. due to
anthropogenic modifications of its morphology), a consolidation/desagregation
module of the bottom must be considered. In fact, after deposition, sediments are
submitted to a consolidation process resulting into an increase of the critical shear
stress for erosion. To simulate the erosion process an initial vertical profile of
consolidation has to be specified.
3.3.3 Boundary conditions
Boundary conditions can be specified in terms of a specified value, a specified flux or
a specified law of property variation at the boundary. The radiation boundary
conditions used on ocean hydrodynamic models fit in the latter group. A typical
coastal lagoon has one or several sea inlets and receives land discharges through one
or several rivers. In general, at sea inlets the most convenient is to specify the values
of the properties or to estimate those values as a function of the values inside the
modelling area. At the river boundary the flux is the river flow rate times the specific
value of the property (concentration in case of a mass). In this case properties in the
fresh water are the most convenient boundary condition. When multiplied by the river
discharge the condition becomes in fact a flux condition.
3.3.4 Internal coefficients: calibration and validation
Internal coefficients are used to parameterise empirical closures of the processes
simulated by the model. These parameters have to be fitted using field data. This
procedure is called calibration. From literature the range of variation of each
parameter is known. When calibration procedure suggests parameter values out of that
range, a scientific explanation has to be found. In fact the most common reason for
calibration values out of range is the need to include extra processes in the model.
After calibration the model must be validated using a data set not used in the
calibration process. This validation process will guarantee that the parameters are
adequate to a range of conditions representative of those found in the system being
studied.
The calibration effort increases with in a non-linear way with the number of
parameters (e.g. biological models). In case of models with simple spatial grids the
running time is small and some automatic procedures can be established to select the
best values of the parameters. In practice, in real systems horizontal transport
simulation is required and a small number of trials can be done.
3.4
Model analysis (b)
3.4.1 Limits of models (b)
3.4.1.1 Limitations due to the physical approach
3.4.1.2 Limitations due to the numerical scheme
3.4.1.3 Limitations due to boundary conditions
3.4.1.4 Limitations due to internal parameters
3.4.1.5 General approach to minimize model limitations
3.4.2 Sensitivity Analysis
3.4.3 Calibration
3.4.4 Validation
3.5
User interface (r+f)
List of Figures
Figure 3.5-1: Example of a grid for a 3D computation. Two vertical domains are
used. In the upper domain, a sigma coordinate is used while lower
one uses a Cartesian.
Figure 3.5-2: Typical 1D spatial grid.
Figure 3.5-3: Generic control volume in a 1D discretisation.
Figure 3.5-4: Visualization of the consequences of temporal discretisation.
Property evolves within a time step, but not values used for flux
calculation.
Figure 3.5-5:
List of equations
Equation 3.5-1:
Difference equation for a generic 1D transport equation.
Equation 3.5-2:
Difference 1D transport equation for the particular case of
uniform and permanent geometry.
Equation 3.5-3:
Central-difference calculation of advection.
Equation 3.5-4:
Upstream calculation of advection.
Equation 3.5-5:
QUICK calculation of advection.
Equation 3.5-6:
Advection equation discretised using central explicit
differences.
Equation 3.5-7:
Advection-diffusion equation discretised using central
explicit differences for advection and for diffusion.
Equation 3.5-8:
Conditions for numerical stability of 1D explicit advectiondiffusion discretised using central differences.
Equation 3.5-9:
Advection equation discretised using upstream explicit
differences.
Equation 10: Finite difference for computing temporal derivative in an explicit
method.
Equation 11: Finite difference for computing temporal derivative in an implicit
method.
Equation 12: Taylor expansions for deriving semi-implicit methods.
Equation 13: Finite difference for computing temporal derivative in an CrankNicholson (semi-implicit) method.
Equation 14: Expansion series for computing C at point “i+1”
Equation 15: Expansion series for computing C at point “i-1”
Equation 16: Expansion series for computing first spatial derivative using
central differences.
Equation 17: Expansion series for computing first spatial derivative using left
differences.
Equation 18: Expansion series for computing first spatial derivative using right
differences.
Equation 19: Expansion series for computing second spatial derivative using
central differences.
List of Tables
Table 3.5-1: Example of a time evolution in a 1D channel computed using explicit
central differences, a unitary courant number, and no diffusion.
Table 3.5-2: Example of a time evolution in a 1D channel computed using explicit
central differences, Cr=2, and no diffusion.
Table 3.5-3: Example of a time evolution in a 1D channel computed using explicit
upstream differences, Cr=1.0, and no diffusion.
Table 3.5-4: Example of a time evolution in a 1D channel computed using explicit
upstream differences, Cr=0.5, and no diffusion.