Numerical Advection Schemes in Two Dimensions

University of Exeter
ECMM713: Modelling Applications and
Case Studies
Numerical Advection Schemes
in Two Dimensions
Author:
Hugo Winter
Supervisor:
Prof. John Thuburn
April 26, 2011

1 INTRODUCTION
1 Introduction
Computational models lie at the heart of forecasting in the 21st century.
The process of forecasting is much more than numerical weather prediction
alone, yet without it the whole system would collapse. With these models
it has become possible to predict on wide ranging time-scales, from the
weather outside tomorrow to the level of greenhouse gases in the atmosphere
in hundreds of years time. Such important applications have necessitated
the development of more and more complicated and ingenious schemes to
try and model the dynamics of the climate system. It is this development
that has progressed numerical weather prediction forwards which makes this
a significant and interesting area to research.
The science of fluid dynamics revolves around the idea of fluid particles
being transported in time and space. This can occur in many types of media
from water to air and is known as advection. This process is very important
when understanding the atmosphere and as such necessary when attempting
to predict the weather (Holton 2004). Without a good understanding of the
advective processes at work it would be near impossible to give a prediction
on the weather a day in advance, let alone a week in advance. Due to the
increase in computing power in recent years it has become possible to create
complicated algorithms which model this advection process. As such, many
different numerical advection algorithms have sprung up giving much scope
for model choice (Rood 1987). Many of these models have originated in
different branches of science, some have come from plasma physics others
from oceanography, but most can be used to model the advective processes
in the atmosphere.
With such choice on offer it therefore becomes paramount to choose the
correct algorithm for the situation. Factors such as computing cost and
accuracy needed have to be taken into account. Problems with the stability
of certain schemes also contribute to their utility. There is little point in
running a global climate model for several years if the results have blown
up after a week. On the other hand, an accurate and stable scheme that
takes five days to predict the weather tomorrow is also of little use. Many of
the so-called ‘classical’ advection schemes work well on standard grids with
constant velocity of the flow, but may not be able to model more complicated
features of the flow. On the other hand more complicated schemes that take
into account these factors may simply be too expensive for all but a large
supercomputer (Staniforth and Côte 1991).
The aim of this report is to look at the two-dimensional (2D) advection
equation. Two-dimensional fields produce interesting problems for advec-
1

2 THE ADVECTION EQUATION
tion schemes. Stability properties change compared to the one-dimensional
equation and there is the possibility of advecting at different speeds in different
directions. Certain schemes can be directly extended from the onedimensional
(1D) scheme, whereas others might require each step to be split
in two orthogonal directions (LeVeque 2002).
2 The advection equation
2.1 Deriving the advection equation
In a fixed volume of fluid which does not contain any sources or sinks, the
rate of change of a constituent is equal to the amount transported in and
out of the fixed volume (Rood 1987). This gives the constituent continuity
equation
DC
Dt ≡ ∂C
∂t + u · ▽C
Another important quantity is the mixing ratio. It is defined below
ϕ ≡ C ρ
When the two equations above are combined, we can rewrite the continuity
equation as,
Dϕ
Dt = ∂ϕ
( )
∂ρ
∂t + u · ▽ϕ = −ϕ ρ ∂t + ▽ · ρu
However, now the right hand side of the continuity equation is just the mass
continuity equation for the fluid which is equal to zero so finally the advection
equation can be written
∂ϕ
∂t + u · ▽ϕ = 0
For the purpose of this report we are going to work with the two-dimensional
advection equation. If we take this case and assume the velocity of the flow
in the x and y directions to be positive and constant the above equation can
be rewritten
∂ϕ
∂t + u∂ϕ ∂x + v ∂ϕ
∂y = 0 (1)
2

2.2 Taylor Series 2 THE ADVECTION EQUATION
For this equation an analytical solution is known. Having an analytical solution
allows us to see how well our transport scheme is doing in comparison.
If a scheme has good properties it can then be extended to more complicated
cases. To create our different schemes for solving the advection equation it
is first important to define a Taylor Series.
2.2 Taylor Series
The Taylor series expansion provides a method with which the advection
equation can be solved computationally. The point at (i + 1, j) can be expressed
as a Taylor Series
ϕ i+1,j = ϕ i,j + ∆x dϕ i,j
dx + ∆x2 d 2 ϕ i,j
+ O(∆x 3 ) (2)
2 dx 2
This Taylor series has been expanded around the point (i, j) although it can
be used to expand around any point. The point at (i − 1, j) can also be
written as a Taylor Series expanded about the point (i, j)
ϕ i−1,j = ϕ i,j − ∆x dϕ i,j
dx + ∆x2 d 2 ϕ i,j
− O(∆x 3 ) (3)
2 dx 2
The O(∆x n ) term at the end of the expression is called the truncation error
of our finite difference approximation (Tu et al. 2008). Ideally we would
like the truncation error to be of as small order as possible. However, it is
important to balance out the added accuracy with the extra complexity that
the additional terms provide.
Two fundamental schemes are the forward difference scheme (FTBS) and
the centred difference scheme (CTCS). Both of these schemes have benefits
and drawbacks in terms of errors and stability. These schemes are cheap
in terms of computational cost, however FTBS is diffusive and CTCS is
dispersive (discussed in section 2.3) and both produce large truncation errors.
Using the Taylor expansion (2) (expanded with respect to y and t also) it is
possible to rewrite the advection equation (1) to obtain the forward difference
scheme
ϕ m+1
i,j
− ϕ m i,j
∆t
= −u ϕm i+1,j − ϕ m i,j
∆x
Which can be rearranged to give
− v ϕm i,j+1 − ϕ m i,j
∆y
+ O(∆x, ∆y, ∆t)
3

2.3 Errors and Stability 2 THE ADVECTION EQUATION
ϕ m+1
i,j = ϕ m i,j − U(ϕ m i+1,j − ϕ m i,j) − V (ϕ m i,j+1 − ϕ m i,j) + O(∆x, ∆y, ∆t) (4)
Where U and V are Courant numbers which are defined as
U = u ∆x
∆t ,
V = v ∆y
∆t
From the definition of the scheme in equation (4), it can be seen that the
scheme has first order error. The scheme only requires the points upwind of
(i, j) to make the step and does not require information from any previous
time steps. By taking equation (1) minus equation (2) we can then obtain
the centred difference scheme
ϕ m+1
i,j
− ϕ m−1
i,j
2∆t
= −u ϕm i+1,j − ϕ m i−1,j
2∆x
− v ϕm i,j+1 − ϕ m i,j−1
2∆y
+ O(∆x 2 , ∆y 2 , ∆t 2 )
Which can be rewritten as
ϕ m+1
i,j
= ϕ m−1
i,j − U(ϕ m i+1,j − ϕ m i−1,j) − V (ϕ m i,j+1 − ϕ m i,j−1) + O(∆x 2 , ∆y 2 , ∆t 2 ) (5)
Where the Courant numbers are defined in the same way as above. This
scheme has an error of the second order which is an improvement on the
forward difference scheme. To obtain this extra accuracy, points either side
of (i, j) and values from the previous time step are also required. When using
an advection scheme we want the errors to be as small as possible. However,
for finite difference methods the validity of the model is determined more by
its stability, a process closely linked to the Courant number defined above.
2.3 Errors and Stability
When modelling an advective process it is important to know the type of
errors that might occur. A perfect numerical advection scheme would transport
a fluid particle in space, returning it to exactly the same place that it
started. However when truncating the Taylor Series errors are introduced
into the system. Roughly these errors split into two different types, dispersion
and diffusion (Rood 1987). Dispersion errors occur when a scheme
introduces small-scale waves, as different Fourier components propagate at
4

3 ADVECTING IN TWO DIMENSIONS
different phase speeds. These type of errors are most commonly seen on centred
difference schemes. Forward differencing suffers from diffusion errors,
which are problematic when dealing with shock fronts. Any rectangular distributions
will be smoothed and eventually all types of distribution will be
dissipated across the field. Many of the ‘classical’ numerical methods (such
as the ones discussed in this report) are designed to reduce one of these two
types of error. Modern schemes attempt to use a combination of methods
when solutions become too dispersive or to diffusive (Van Leer 1974).
It is also vital for a numerical advection scheme to be stable. An unstable
scheme is of little use, if the solution is not bounded as time goes to infinity we
cannot have confidence that it will produce accurate results for any useful
length of time. The Courant number defined earlier usually needs to be
bounded to within a certain set of values for a scheme to be stable. Using
the example of FTBS again, in one dimension it turns out that for the scheme
to be stable the Courant number U needs to be bounded 0 < U < 1.
In two dimensions the calculations are slightly more complicated. The
Courant numbers U and V may not be the same and the grid spacing could
be different in the x and y directions. Even if the grid spacing is the same,
not all 1D advection schemes can be simply extended into two dimensions.
To investigate the stability we shall use Von Neumann stability analysis. For
this type of analysis to work the scheme has to be linear and the coefficients
must be constant. These conditions are satisfied if the advecting velocities u
and v are set to be constants.
3 Advecting in two dimensions
3.1 Direct extension
In one dimension, for the centred difference scheme to be stable it turns
out that the Courant number is bounded by the value 1 from above. In
two dimensions it is possible to extend this scheme directly from the onedimensional
case, although the values for the Courant number for which the
scheme is stable will change. As mentioned above we shall use Von Neumann
stability and this requires us to seek solutions of the form
ϕ m i,j = A m exp {i(kn∆x + lj∆y)} (6)
Where k and l are the wave numbers in the x and y directions respectively.
5

3.1 Direct extension 3 ADVECTING IN TWO DIMENSIONS
Now substituting equation (6) into equation (7) we obtain after some rearrangement
A = −i (Usin(k∆x) + V sin(l∆y)) ± [1 − (Usin(k∆x) + V sin(l∆y))] 1/2
For stability we need that |A| ≤ 1. This holds if
|Usin(k∆x) + V sin(l∆y)| ≤ 1 (7)
At this point we shall consider the simple case of a square grid ∆x = ∆y = q.
This permits the velocities u and v to be defined in terms of a specific flow
speed (f s )
In this way equation (7) becomes
u = f s cos(θ), v = f s sin(θ) (8)
f s ∆t
|cos(θ)sin(kq) + sin(θ)sin(lq)| ≤ 1 (9)
q
For this to be satisfied for all waves in the 2D plane it serves to prove it for
the most rigorous case, when sin(kq) = sin(lq) = 1. This reduces equation
(10) to
f s ∆t
|cos(θ) + sin(θ)| ≤ 1 (10)
q
We wish to look at the limiting case for which the flow is moving diagonally.
This implies that θ = π 4 and the value of |cos(θ) + sin(θ)| = √ 2. So for the
two-dimensional case where we are on a square grid box and u and v are
advecting with the same flow speed, this implies that
f s ∆t
q
≤ 1 √
2
(11)
The explanation behind this can be seen better in figure 1. With CTCS a
wave signal cannot propagate more than one grid interval without becoming
unstable. When combined with equations (8) and (9) the conditions on U
and V are
U ≤ 0.5, V ≤ 0.5 (12)
6

3.2 Directional splitting 3 ADVECTING IN TWO DIMENSIONS
0.707d
d
3.2 Directional splitting
Figure 1: 2D wave propagation
It is not possible to extend all numerical advection methods into two dimensions
quite so easily. An interesting example is the Lax-Wendroff method.
In one dimension the method is defined as
ϕ m+1
i = ϕ m i − U 2 (ϕm i+1 − ϕ m i−1) + U 2
2 (ϕm i+1 − 2ϕ m i + ϕ m i−1) + O(∆x 2 , ∆t 2 )
where U is the 1D Courant number u ∆t . For this case the Courant number is
∆x
stable between −1 and 1. However if we just add on the additional terms for
the y-direction the scheme becomes completely unstable. The explanation
for this lies within the derivation of the Lax-Wendroff method. The starting
point for the derivation is the Taylor series of ϕ about t
∂ϕ(x, y, t)
ϕ(x, y, t ± ∆t) = ϕ(x, y, t) ± ∆t + ∆t2 ∂ 2 ϕ(x, y, t)
+ ... (13)
∂t 2 ∂t 2
Using equation (1) each of the derivatives with respect to t can be written in
terms of x and y. For notational simplicity we shall use the convention that
can be rewritten ϕ t . As such we have that
∂ϕ(x,y,t)
∂t
ϕ t = −uϕ x − vϕ y (14)
ϕ tt = −u 2 ϕ xx − v 2 ϕ yy + 2uvϕ xy (15)
Substituting in the formulae for the first and second derivatives of secondorder
centred differences (Rood 1987)
7

3.2 Directional splitting 3 ADVECTING IN TWO DIMENSIONS
ϕ x = ϕ i+1,j − ϕ i−1,j
2∆x
we obtain the fully 2D Lax-Wendroff scheme
(16)
ϕ xx = ϕ i+1,j − 2ϕ i,j + ϕ i−1,j
∆x 2 (17)
ϕ m+1
i,j = ϕ m i,j − U 2 (ϕm i+1,j − ϕ m i−1,j) − V 2 (ϕm i,j+1 − ϕ m i,j+1)
− U 2
2 (ϕm i+1,j − 2ϕ m i,j + ϕ m i−1,j) − V 2
2 (ϕm i,j+1 − 2ϕ m i,j + ϕ m i,j−1)
+ UV 4 (ϕm i+1,j+1 − ϕ m i−1,j+1 − ϕ m i+1,j−1 + ϕ m i−1,j−1)
As can be seen from the above equation there is a cross derivative term in
the fully 2D Lax-Wendroff scheme. However, if the 1D scheme is simply
extended this term is completely omitted, hence the direct extension method
becomes completely unstable. One method that can be used to remedy this
problem is to use directional splitting. This method consists of breaking
a finite-difference formula into a series of steps (Durran 1999). In the 2D
problem that we have been using, it corresponds to taking the x step first
followed by a y step. So the fully 2D Lax-Wendroff equation can be rewritten,
using differential operator notation (see appendix I), as
(ϕ m+1
i,j ) ∗ = ϕ m i,j − u∆tδ 2x ϕ m i,j + (u∆t)2 δ xx ϕ m i,j (18)
2
ϕ m+1
i,j = (ϕ m i,j) ∗ − v∆tδ 2y (ϕ m i,j) ∗ + (v∆t)2 δ yy (ϕ m
2
i,j) ∗ (19)
The main benefit of this method is that fewer computations are required
to solve the split system, thus reducing computational cost. It also relies
on the only the 1D form of Lax-Wendroff which is a scheme that is well
understood. The stability restrictions on the Courant number are also much
less severe than for fully 2D schemes. Here, each of the steps is bounded
by the stability condition for the 1D Courant number (−1 ≤ U ≤ 1). One
major drawback is the necessity to choose orthogonal steps for this method
to work. As such we are restricted in the types of grid that can be chosen for
the 2D problem. The choice of steps is interesting and there are two main
options here. Figure 2(a) shows splitting defined in equations (18) and (19).
8

3.3 Fully 2D schemes 3 ADVECTING IN TWO DIMENSIONS
(a) Splitting from equations
(18) and (19)
(b) Strang splitting
Figure 2: Different types of directional splitting
On the other hand, figure 2(b) shows Strang splitting which takes one step
of the previous method and then reverses it for one step. It may be intuitive
that the second method would give a more accurate solution, however the
difference is not always that great (LeVeque 2002). The splitting of the
problem could also lead to some loss of accuracy and spurious error terms of
O(∆t) 2 . It is very difficult to know in advance whether satisfactory results
will be obtained using directional splitting (Kuzmin and Löhner 2005). As
such we have to be careful when using this method, implying that a more
robust fully 2D scheme is required.
3.3 Fully 2D schemes
The methods above have aimed to take schemes and extend them using
the 1D form of the advection equation. There are benefits to this, the 1D
advection equation is well known and many of the methods are relatively
simple. On the other hand there will be some loss in accuracy since we are
trying to use 1D schemes to approximate what is a fully 2D process. As
such it would be beneficial to have a fully 2D scheme to use, which should
be the most accurate scheme that we can create. In the previous section
we defined the 2D Lax-Wendroff scheme to illustrate the cross-derivative
term that appears. Implementing this directly should hopefully give the
most accurate representation and reduce the errors to their minimum. The
stability condition for the fully 2D method is defined as
U 2/3 + V 2/3 ≤ 1 (20)
9

4 SIMULATION STUDY
This condition is very restrictive, much more so than that for CTCS and the
directional split Lax-Wendroff method. It is possible to improve the stability
if a different approximation to the mixed spatial derivative term in the fully
2D Lax-Wendroff scheme is used. If u ≥ 0 and v ≥ 0 we must replace the
cross-derivative term (Durran 1999)
with the term
UV ∆tδ 2x δ 2y ϕ m i,j (21)
UV ∆tδ x δ y ϕ m i− 1 2 ,j− 1 2
(22)
The change in the term means that the fully 2D Lax-Wendroff scheme is
stable for 0 ≤ U ≤ 1 and 0 ≤ V ≤ 1. This adaptation makes the scheme
more flexible and as such is the best scheme to illustrate the fully 2D Lax-
Wendroff scheme during my simulation study.
4 Simulation study
Each of the methods which have been set out so far have their strengths and
weaknesses. To test which of these methods is better I have set up a simulation
study, which has been undertaken using the mathematical program
Matlab. Three schemes shall be under scrutiny in this section, CTCS, Lax-
Wendroff with directional splitting and the fully 2D Lax-Wendroff scheme.
For the directional split I have used the methodology outlined in equations
(18) and (19) since there is no difference between the results obtained by this
method and the Strang splitting. Intuition suggests that both Lax-Wendroff
schemes should be superior to CTCS, with the added robustness of the 2D
Lax-Wendroff scheme making it the best for all situations.
The study will consist of different initial conditions being set up on a box
of size 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Within this box it is possible to set up a grid
mesh, for the purpose of this study I shall keep to a square grid mesh. Each
scheme will then be used to transport the initial condition around the box
and back to the original place that it started from. At this point it is possible
to quantify the error in terms of the deviation from the initial condition. A
perfect scheme would bring the initial condition back around to the same
place after any number of revolutions. The 2-norm and infinity norm are two
ways to quantify the error and are defined as
10

4 SIMULATION STUDY
ϵ 2 =
( 1
N ΣN i=1Σ N j=1 | ϕ num
i,j
ϵ ∞ = Max ∀i,j | ϕ num
i,j
− ϕ ex
i,j | 2 ) 1/2
(23)
− ϕ ex
i,j | (24)
where ϕ num
i,j is the numerical solution and ϕ ex
i,j is the exact solution. The
2-norm takes into account the errors from all over the plane whereas the
infinity norm is only concerned with the one point where the largest error
occurred. To build an accurate picture of the accuracy of each method it
is important to provide both types of error. One scheme may create large
errors on the wave being advected but small errors across the rest of the plane
which leads to a small 2-norm, however this scheme may not necessarily be
any better.
The simulation will also take into account two different types of initial
condition. A curved wave such as that created by a Gaussian distribution will
be the first type tested and it would be expected that the errors for this initial
condition would be relatively small. There are no sudden changes from one
grid point to another which can strongly affect the error. Secondly advecting
a square wave will test how well each of the schemes deal with shock fronts.
These type of waves can be the most difficult schemes to advect accurately,
especially for even order schemes such as the ones being tested since they are
dispersive.
It is important to note that during this study the flow will be advecting
with a constant velocity in both directions. All the equations that have been
derived up until this point are done so with this assumption in mind. The
grid mesh will also be set to a 100x100 grid so that the Courant number can
only be varied by increasing or decreasing the flow velocity or time step.
Table 1: Error results for Gaussian wave U = V = 0.5
2 − Error ∞ − Error
CTCS 0.0038 0.0214
DSLW 0.0042 0.0274
2DLW 0.0054 0.0343
The error results for the Gaussian wave are very interesting since they
seem to suggest that the CTCS scheme is best when it comes to reducing the
error. CTCS is a dispersive scheme with no diffusion, but the diffusive nature
of the wave being advected helps to reduce the error. The error results for the
11

4 SIMULATION STUDY
1
1.5
1
1.2
0.9
0.8
1
0.9
0.8
1
0.7
0.7
0.8
0.6
0.5
0.6
0.6
y
0.5
y
0.5
0.4
0.4
0.3
0
0.4
0.3
0.2
0.2
0.1
−0.5
0.2
0.1
0
−0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
(a) CTCS
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
(b) Directional split Lax-Wendroff
1
1.2
1
1
0.9
1
0.9
0.9
0.8
0.7
0.8
0.8
0.7
0.8
0.7
0.6
0.6
0.6
0.6
y
0.5
0.4
y
0.5
0.5
0.4
0.3
0.2
0.4
0.3
0.4
0.3
0.2
0
0.2
0.2
0.1
−0.2
0.1
0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0
(c) Fully 2D Lax-Wendroff
(d) True Solution
Figure 3: Square wave advected once around plane U = V = 0.5
Lax-Wendroff schemes are close to that of CTCS, but introduce slightly more
error. The directional split Lax-Wendroff scheme also appears to be reducing
the error better than the fully 2D Lax-Wendroff. This is a surprising result
since it would be expected that a fully 2D scheme would work better when
advecting in two dimensions.
The results obtained in table 2 and figure 3 show what occurs for a square
wave. The wave has again been advected once around the plane and with
Courant numbers U = V = 0.5. The dispersive CTCS scheme creates large
errors when transporting a square wave since the scheme is quickly dominated
by shortwaves.The diffusion of the Lax-Wendroff method controls this
dispersive effect. For this reason the Lax-Wendroff method is far superior
to the CTCS for carrying the square wave. It is again significant that the
directional split Lax-Wendroff method produces less error than the fully 2D
Lax-Wendroff. This pattern suggests that the tests being performed with
constant flow velocities do not create any spurious errors in the directional
12

4 SIMULATION STUDY
0.2
1.5
0.18
0.16
CTCS
TSLW
2DLW
CTCS
TSLW
2DLW
0.14
1
2−Error
0.12
0.1
0.08
Ininity−Error
0.06
0.5
0.04
0.02
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Courant Number
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Courant Number
(a) 2-error
(b) ∞-error
Figure 4: How errors vary with courant number
split method. In 3(a), peaks and troughs between −0.5 and 0.5 in the field
away from the initial condition (figure 3(d)) show that CTCS is a more dispersive
scheme. This result compares to peaks and troughs of just between
−0.2 and 0.2 for the Lax-Wendroff methods. Figures 3(b) and 3(c) suggest
that the square wave is being advected in a similar manner by both Lax-
Wendroff schemes, but it is the rest of the field that is having an effect on
the error values. The fully 2D scheme seems to produce more and slightly
stronger dispersive waves in its wake, whereas the dispersion created by the
directional split Lax-Wendroff is weaker.
So far we have set the Courant number to a certain rate and looked
at how the different schemes perform. A situation may occur where for
certain Courant numbers, different schemes may be better. As such figure
4 looks at how the error varies with different Courant numbers. For this
graph the square wave from figure 3 was chosen with U = V . The graphs
show that the directional split Lax-Wendroff method reduces the error for all
Courant numbers compared to its fully two dimensional counterpart. The ∞-
error of the CTCS is smaller for certain Courant numbers than the fully 2D
Lax-Wendroff scheme. Physically, the change in Courant number represents
the size of the time-step being taken (when the grid spacing and velocities
have been fixed). The reduction in time step improves the CTCS greatly
since it requires three-time levels. Conversely, as the time-step is increased,
the undamped computational mode of CTCS generates instability (timesplitting).
Figure 4 shows that stability becomes a problem when U and V
are greater that 0.5 (equation (12)).
13

5 CONCLUSION
5 Conclusion
Table 2: Error results for square wave U = V = 0.5
2 − Error ∞ − Error
CTCS 0.1977 1.3888
DSLW 0.1073 0.7911
2DLW 0.1170 0.8050
The aim of this project was to take the well known one-dimensional advection
equation and extend it into two-dimensions. By comparing three different
second order schemes, the different processes behind model extension have
been discussed. Schemes such as CTCS can be extended straight from their
1D counterparts and remain stable, although under restricted stability conditions.
The Lax-Wendroff scheme on the other hand is an example of a
scheme that becomes totally unstable if the 1D case is simply extended. To
overcome this it is either necessary to split directionally or derive a fully twodimensional
scheme. These schemes have the benefit of less strict stability
conditions which make them more useful.
To test the theory, Matlab was used to construct simulations of an advecting
fluid. Analysis of the errors produced by each of the different schemes
allows the accuracy of each of them to be quantified. For a Gaussian wave it
turned out that CTCS was a very good scheme to use which performed better
than both of the Lax-Wendroff methods. On the other hand, for square
waves the small amount of dissipation added by the Lax-Wendroff scheme
made it a better scheme to use. One interesting result was the fact that
the directional split scheme worked better than the fully two-dimensional
scheme. An extension to this study would be to see if this holds for nonconstant
advecting velocities. I would expect that the fully 2D scheme would
be better when advecting with these non-constant velocities or maybe with
different grid shapes.
The application of advection schemes within numerical weather prediction
continues to be very important to modern weather forecasting. If one thing
has become apparent over this project is that there is such a large amount
of schemes available for the user. The schemes that I have studied here may
not be the ones that are being used for modern applications, but they are
still important. They highlight where numerical weather prediction has come
from and the rapid growth suggests that it can improve many years into the
future.
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Appendix I
Finite-difference operator notation is used to write formulae in compact form.
Necessary combinations of x and y directions can be made from the following
(Durran 1999)
References
δ 2x ϕ m i,j = ϕm i+1,j − ϕ m i−1,j
2∆x
δ 2 xϕ m i,j = ϕm i+1,j − 2ϕ m i,j − ϕ m i−1,j
(∆x) 2
[1] Durran, D.R., Numerical Methods for Wave Equations in Geophysical
Fluid Dynamics, Springer, 1999.
[2] Holton, J.R., An introduction to dynamic meteorology, 4th ed., Elsevier,
2004.
[3] Kuzmin, D., Löhner, R., Flux-corrected transport: principles, algorithms,
and applications, Springer, 2005.
[4] LeVeque, R.J., Finite volume methods for hyperbolic problems, Cambridge,
2002.
[5] Rood, R.B., 1987, Numerical Advection Algorithms and Their Role in
Atmospheric Transport and Chemistry Models, Review of Geophysics,
25, 71-100.
[6] Staniforth, A. and Côte, J., 1991, Semi-Lagrangian Integration Schemes
for Atmospheric Models - A Review, Monthly Weather Review, 119, 2206-
2223.
[7] Tu, J., Yeoh, G.H. and Liu., C., Computational Fluid Dynamics: A Practical
Approach, Butterworth-Heinemann, 2008.
[8] Van Leer, B., 1974, Towards the Ultimate Conservative Difference
Scheme. II. Monotonicity and Conservation Combined in a Second-Order
Scheme, Journal of Computational Physics, 14, 361-370.
15

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REFERENCES
Appendix II
Matlab code used for simulation study in section 4.
% Domain size (periodic)
l = 1.0;
% Number of grid points in domain
n = 100;
% Grid
dx = l/n;
dy = 1/n;
[x,y] = meshgrid(linspace(dx,l,n),linspace(dy,l,n));
% Advecting velocity
u = 1.0;
v = 1.0;
% Time step
dt = input(’Timestep > ’);
% Courant number
cu = u*dt/dx
cv = v*dt/dy
% Number of steps to take in order to go
% once around
nstop = ceil(n/cu);
for k = 1:n
for j = 1:n
xx = x(k,j);
yy = y(k,j);
% Gaussian wave initial condition
f(k,j) = (1/(2*pi*0.4*0.4))*exp(-(1/2)*
(((xx-0.5)^2)/(0.1)^2 +
((yy-0.5)^2)/(0.1)^2));
% Square wave initial condition
if (xx+yy >= 0.4 & xx+yy < 0.8 &
yy >= xx-0.3 & yy < 0.3+xx)
f(k,j) = 1.0;
else
f(k,j) = 0.0;
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end
end
end
h = f;
contourf(x,y,f)
scheme = input(’ 1. CTCS \n 2. Lax-Wendroff \n 3. Lax-Wendroff
with time splitting \n 4. Alternate split LW \n
5. Fully 2D LW \n Which scheme? ’);
% Is user ready?hh
ready = input(’ Ready ? ’);
% Set up arrays of indices k, k-1 and k+1,
% but allowing for wrap-around (periodic domain)
k = 1:n;
km = k - 1;
km(1) = n;
kp = k + 1;
kp(n) = 1;
p = 1:n;
pm = p - 1;
pm(1) = n;
pp = p + 1;
pp(n) = 1;
% One step needed if using CTCS
if scheme == 1
d(k,p) = f(k,p);
end
% Loop over time steps
for istep = 1:nstop
% CTCS
if scheme == 1
g(k,p) = d(k,p) - cu * (f(kp,p) - f(km,p))
- cv * (f(k,pp) - f(k,pm));
d(k,p) = f(k,p);
f(k,p) = g(k,p);
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end
% Lax-Wendroff scheme
if scheme == 2
f(k,p) = f(k,p) - (cu/2)*(f(kp,p) - f(km,p)) +
((cu^2)/2)*(f(kp,p)-(2*f(k,p))+f(km,p))
-(cv/2)*(f(k,pp) - f(k,pm)) +
((cv^2)/2)*(f(k,pp)-2*f(k,p)+f(k,pm));
end
% Splitting for Lax-Wendroff
if scheme == 3
f(k,p) = f(k,p) - (cu/2)*(f(kp,p) - f(km,p)) +
((cu^2)/2)*(f(kp,p)-2*f(k,p)+f(km,p));
f(k,p) = f(k,p) - (cv/2)*(f(k,pp) - f(k,pm)) +
((cv^2)/2)*(f(k,pp)-2*f(k,p)+f(k,pm));
end
% Strang split
if scheme == 4
if mod(istep,2)==0
f(k,p) = f(k,p) - (cu/2)*(f(kp,p) - f(km,p)) +
((cu^2)/2)*(f(kp,p)-2*f(k,p)+f(km,p));
f(k,p) = f(k,p) - (cv/2)*(f(k,pp) - f(k,pm)) +
((cv^2)/2)*(f(k,pp)-2*f(k,p)+f(k,pm));
else
f(k,p) = f(k,p) - (cv/2)*(f(k,pp) - f(k,pm)) +
((cv^2)/2)*(f(k,pp)-2*f(k,p)+f(k,pm));
f(k,p) = f(k,p) - (cu/2)*(f(kp,p) - f(km,p)) +
((cu^2)/2)*(f(kp,p)-2*f(k,p)+f(km,p));
end
end
% Fully 2D LW
if scheme == 5
fiplusm(k,p) = (1/2)*(f(k,p)+f(kp,p)) -
(1/2)*cu*(f(kp,p)-f(k,p));
fjplusm(k,p) = (1/2)*(f(k,p)+f(k,pp)) -
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(1/2)*cv*(f(k,pp)-f(k,p));
fiminusm(k,p) = (1/2)*(f(km,p)+f(k,p)) -
(1/2)*cu*(f(k,p)-f(km,p));
fjminusm(k,p) = (1/2)*(f(k,pm)+f(k,p)) -
(1/2)*cv*(f(k,p)-f(k,pm));
end
f(k,p) = f(k,p) - cu*(fiplusm(k,p)-fiminusm(k,p)) -
cv*(fjplusm(k,p)-fjminusm(k,p)) +
cu*cv*(f(k,p)-f(k,pm)-f(km,p)+f(km,pm));
% Plot f at this timestep
contourf(x,y,f)
pause(0.1)
end
err = abs(f - h);
rss = sum(sum(err.^2));
rss3 = sum(sum(err.^3));
error2 = (rss/(n^2))^(1/2);
error3 = (rss3/(n^2))^(1/3);
errorinf = max(max(err));
errorstuff = [error2;error3;errorinf]
contourf(x,y,f);
xlabel(’x’)
ylabel(’y’)
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