Using Partitioning in the Numerical Treatment of
ODE Systems with Applications to Atmospheric
Modelling
Zahari Zlatev
National Environmental Research Institute
Frederiksborgvej 399, P. O. Box 358
DK-4000 Roskilde, Denmark
e-mail: zz@dmu.dk
http://www.dmu.dk/AtmosphericEnvironment/staff/zlatev.htm
http://www.dmu.dk/AtmosphericEnvironment/DEM
http://www.mathpreprints.com
Contents
1. Major assumptions
2. Partitioning into strong and week blocks
3. Advantages of the partitioned algorithms
4. “Local” error of the partitioned algorithms
5. “Global” error of the partitioned algorithms
6. Application to some atmospheric models
7. Computing times
8. Open problems
Great environmental challenges
in the 21st century
1. More detailed information about the pollution
levels and the possible damaging effects
2. More reliable information (especially about
worst cases)
How to resolve these two tasks?
By using large-scale mathematical models
discretized on a fine resolution grids
Mathematical Models
 cs
(uc s ) (vcs )


t
x
y
 cs 
   cs   
   K y

  K x
x 
 x  y 
y 
 (k1s  k 2 s )c s
 Es  Qs(c1,c2 ,... ,cq )
(wcs )    cs 


  K z
z
z 
z 
s  1,2 ,...,q
hor. transport
hor. diffusion
deposition
chem.  emis.
vert. transport
“Non-optimized” code
Module
Comp. time
Percent
Chemistry
16147
83.09
Advection
3013
15.51
Initialization
1
0.00
Input operations
50
0.26
Output operation
220
1.13
Total
19432
100.00
It is important to optimize the chemical part for this problem
2-D version on a 96x96 grid (50 km x 50 km)
The time-period is one month
The situation changes for the fine resolution models (the
advection becomes very important)
The computing time is measured in seconds
One processor is used in this run
Methods for the chemical part
QSSA (Quasi-Steady-State-Approximation)
 Classical numerical methods for stiff ODEs
 Partitionig

QSSA
dy
 P (y , y ,..., y )  L (y , y ,..., y ) y
dt
s
s
P
y 
L
n 1
1
2
q
s
2
for
s
s
q
s
L t  10
s
s
P 
P
y    y   e
L 
L 
y  y  t P  L y
n 1
1
s
s
n
 t L s
s
s
s
n 1
n
s
s
s
s
s
s
for 0.01  L t  10
s

for
L t  0.01
s
1. Major assumptions
It will be assumed that the components of
the solution vector can be divided into several
groups, so that the components belonging to
different groups have different properties.
In the particular case where the chemical part of
a large air pollution models is studied, the
reactions are divided into fast reactions and slow
reactions
yn
2. Partitioning vector
dy
 f (t , y ), y   q , f   q
dt
yn 1  yn   t f n 1 ,
yn  y (t n ), f n  f (t n , yn )
I  tJ y
[0]
n 1
Assume
yn is
[i ]
n 1
  yn[ i11]  yn  t f (t n 1 , yn[i11] )
that :
partitione d to NBLOCKS
blocks
r1 , r2 , ... , rp  p  q, 1  rk  q, k  1, 2, ... , q  one of the blocks
The paritionin g of yn implies partitioni ng in
Two row blocks of
Strong
and
I  tJ n[ 0]1
I  tJ n[ 0]1
correspond to each block of
week blocks (the strong
yn
blocks are diagonal)
3. Forming the partitioned system
A y
[0]
n 1
[i ]
n 1
  yn[ i11]  yn  t f (t n 1 , yn[ i11] )
An[ 0]1  I  tJ n[ 0]1
An[ 0]1  S n[ 0]1  Wn[01]
Let
yn[ 1]
obtained
A y
[0]
n 1
[ ]
n 1
~ [ 0]
S n[ 0]1  I  t S n 1
with
be the accepted
by
 y
the regular
[  1]
n 1

solution
Newton method
 yn  t f t n 1 , y

[  1]
n 1
S n[ 0]1z n[ i]1   z n[ i11]  z n  t f t n 1 , z n[ i11]


Splitting matrix A
A=S+W
S W W  S

 
 W S W   0
 W W S  0

 

11
21
0
12
13
22
23
S
33
0
31
32
A
11

S
0  0 W W 

 
0 W 0 W 
S   W W 0 

12
22
13
21
33
23
31
32
W
4. Why partitioned system?
Advantages:
1. It is not necessary to compute the non-zero
elements of the weak blocks
2. Several small matrices are to be factorized
instead of one big matrix
3. Several small systems of linear algebraic
equations are to be solved at each Newton
iteration (instead of one large system)
5. Requirements
A y
[ 0]
n 1
S
z
[0]
n 1
[i ]
n 1
[i ]
n 1
 y
 z
[ i 1]
n 1
[ i 1]
n 1

 t f t
 yn  t f tn 1 , y
 zn
n 1
,z
[ i 1]
n 1
[ i 1]
n 1

It is desirable to find the conditions under which the
approximate solution of the second problem is close
to the approximate solution of the first one
y
[ ]
n 1
z

z
[ ]
n 1
[ ]
n 1
0
is
small
is
not
is not much larger
sufficient


Lemma 1
yn[ 1]  z n[ i]1  Bn[ i11]  yn[ 1]  z n[ i11]   Cn 1 ( yn  z n )  Dn[ i11] yn[ 1]
where
Cn 1  S

[ 0 ] 1
n 1
[ i 1]
n 1
 t S
[ i 1]
n 1
 S
B
D
Tn[i 11] 
1


[ 0 ] 1
n 1

f t
 [ i 1] ~ [ 0 ] 
 Tn 1  S n 1 


[ 0 ] 1
n 1
Wn[01]  Bn[ i11]
[  1]
[ i 1]

y

(
1


)
z
n 1,
n 1
n 1 
yn 1
exact
yn[ 1]
approximat ion
some
d
y
0
solution
numerical
of
of
method
yn 1  yn  t f (t n 1 , yn 1 )
yn 1
found
by
Lemma 2
y
[ ]
n 1
z
[i ]
n 1
 i
 [ ]
[ i  j]
 y n 1  z [n0]1

B

n

1


 j1

 i 1  k

[ i  j]
  
B n 1 
 C n 1 ( y n  z n )





 k  j1

 i 1  k
 [ i 1k ] 
[ i  j]
[ ]

  
B
D

y


n 1 
n 1
n 1




 k  j1


yn 1 exact solution
of
yn[ 1] approximat ion of
some numerical
method

yn 1  yn  t f (t n 1 , yn 1 )
yn 1 found
by
“Local” error
B
n 1
Bn 1  1
Theorem
 max  B
0  j i
[ j]
n 1

Major assumption
[i ]
yn 1  z n
1


[ ]
[i ]
yn
1  z n 1


1. if the number of iterations is sufficiently large
and
2. if the error from the previous step is
sufficiently small
“Global” error
Bk[ k ] 
Ek 
max  B ,
0  k  n 1
[i ]
k
0  i  k
Ck
,
[ k ]
1  Bk
E
max ( E
0  k  n 1
k
)
Theorem
If
Bk[ k ]  1 and
yn1  z
[ n1 ]
n 1

E  1 then the inequalities
and
[ ]
n 1
y z
[ n1 ]
n 1

will be satisfied when sufficiently many iterations
are performed
“Global” error - continuation
If
E  1,
Theorem
yn 1  z n[ vn11 ]  E n 1
y
yn[ 1]  z n[n11 ]  E n 1
y
If
0
0
 z0

 z0

1  E n 1


1 E
1  E n 1


1 E
E  1,
[ v n 1 ]
y n 1  z n
1

y0  z 0
 n
[ n 1 ]
[ ]
yn

z
1
n 1

y0  z 0
 n
Actual partitioning
1. Matrix S, which is obtained after the partitioning is a
block-diagonal matrix containing 23 blocks.
2. The first diagonal block is a 13x13 matrix.
3. The next 22 clocks are 1x1 matrices (a Newton iteration
procedure for scalar equations is used in this part).
4. This partitioning was recommended by the chemists (based
on their knowledge of the chemical reactions)
Variation of the key quantities for different
scenarios
 ( BN )
Scenario
1
2
3
4
5
6
[2.4E-4, 3.3E-2]
[3.7E-4, 2.1E-2]
[5.8E-3, 5.7E-2]
[1.3E-3, 4.3E-2]
[4.3E-3, 2.5E-2]
[5.6E-3, 6.4E-2]
N  1, 2, ... ,1260
 ( Bn 1 )  B n 1 ,
and
E
1.03425
1.02179
1.06040
1.04606
1.02522
1.06882
t  30s
B n 1   ( Bn 1 )  
Variation
Variation of
of the
the key
key quantities
quantities
for
for different
different stepsizes
scenarios
Stepsize
30
10
1
0.1
0.01
Scenario 5
B
2.5E-2
1.4E-2
2.6E-3
3.6E-4
5.3E-5
E
1.025
1.014
1.0026
1.00036
1.000053
Numerical Results
Method
QSSA-1
QSSA-2
Euler
Trapez.
RK-2
Part. dense
Computing time
12.85
11.72
15.11
15.94
28.49
10.09 Based on Euler
Better accuracy with the classical numerical methods
Accuracy results
Numer. Method Scenario 2 Scenario 6
QSSA-1
3.20E-3
4.39E-1
QSSA-2
3.39E-3
3.86E-1
Euler
5.78E-4
3.57E-3
Trapez
5.78E-4
3.57E-3
Part. dense
5.72E-4
3.26E-3
t  30s
The chemical compound is ozone
Conclusions and open problems
1.We have shown that the physical arguments for achieving
successful partitioning can be justified with clear algebraic
requirements.
2. There is a drawback: if the chemical scheme is changed,
then the whole procedure has to be carried out for the new
chemical scheme.
3. Automatic partitioning is desirable.