Using Partitioning in the Numerical Treatment of ODE Systems with Applications to Atmospheric Modelling

Zahari ZlatevZahari Zlatev

National Environmental Research InstituteNational Environmental Research Institute

Frederiksborgvej 399, P. O. Box 358Frederiksborgvej 399, P. O. Box 358

DK-4000 Roskilde, DenmarkDK-4000 Roskilde, Denmark

e-mail: zz@dmu.dke-mail: zz@dmu.dk

http://www.dmu.dk/AtmosphericEnvironment/staff/zlatev.htmhttp://www.dmu.dk/AtmosphericEnvironment/staff/zlatev.htm

http://www.dmu.dk/AtmosphericEnvironment/DEMhttp://www.dmu.dk/AtmosphericEnvironment/DEM

http://www.mathpreprints.comhttp://www.mathpreprints.com

Contents

1. Major assumptions1. Major assumptions

2. Partitioning into strong and week blocks2. Partitioning into strong and week blocks

3. Advantages of the partitioned algorithms3. Advantages of the partitioned algorithms

4. 4. “Local”“Local” error of the partitioned algorithms error of the partitioned algorithms

5. 5. “Global”“Global” error of the partitioned algorithms error of the partitioned algorithms

6. Application to some atmospheric models6. Application to some atmospheric models

7. Computing times7. Computing times

8. Open problems8. Open problems

Great environmental challenges in the 21st century

1. More detailed information about the pollution 1. More detailed information about the pollution levels and the possible damaging effectslevels and the possible damaging effects

2. More reliable information (especially about 2. More reliable information (especially about worst cases)worst cases)

How to resolve these two tasks?How to resolve these two tasks?

By using large-scale mathematical modelsBy using large-scale mathematical models

discretized on a fine resolution gridsdiscretized on a fine resolution grids

Mathematical Models

,...,q,s

z

cK

zz

)(wc

),c...,c(cQE

)ck(k

y

cK

yx

cK

x

y

)(vc

x

)(uc

t

c

sz

s

qss

sss

sy

sx

sss

21

transportvert.

emis.chem.,

deposition

diffusionhor.

transporthor.

21

21

“Non-optimized” codeModuleModule Comp. timeComp. time PercentPercent

Chemistry 16147 Chemistry 16147 83.0983.09

Advection 3013 15.51Advection 3013 15.51

Initialization 1 0.00Initialization 1 0.00

Input operations 50 0.26Input operations 50 0.26

Output operation 220 1.13Output operation 220 1.13

Total 19432 100.00Total 19432 100.00

It is important to optimize the chemical part It is important to optimize the chemical part for this problemfor this problem

2-D version on a 96x96 grid (50 km x 50 km)2-D version on a 96x96 grid (50 km x 50 km)

The time-period is one monthThe time-period is one month

The situation changes for the fine resolution models (theThe situation changes for the fine resolution models (the

advection becomes very important)advection becomes very important)

The computing time is measured in secondsThe computing time is measured in seconds

One processor is used in this runOne processor is used in this run

Methods for the chemical part

QSSAQSSA (Quasi-Steady-State-Approximation) (Quasi-Steady-State-Approximation) Classical numerical methods for Classical numerical methods for stiffstiff ODEs ODEs PartitionigPartitionig

QSSA

01.0

1001.0

10

),...,,(),...,,(

1

1

1

2121

tLforyLPtyy

tLforeL

Py

L

Py

tLforL

Py

yyyyLyyyPdt

dy

ssss

n

s

n

s

s

Lt

s

sn

s

s

sn

s

s

s

sn

s

sqsqs

s

s

1. Major assumptionsIt will be assumed that the components ofIt will be assumed that the components of

the solution vector the solution vector can be divided into severalcan be divided into several

groupsgroups, so that the components belonging to, so that the components belonging to

different groups have different properties.different groups have different properties.

In the particular case where In the particular case where the chemical part ofthe chemical part of

a large air pollution modelsa large air pollution models is studied, the is studied, the

reactions are divided into reactions are divided into fastfast reactions and reactions and slowslow

reactionsreactions

2. Partitioning vector

diagonal)areblocksstrong(theblocksweekandStrong

ofblockeachtocorrespondofblocksrowTwo

inngpartitioniimpliesofgparitioninThe

blockstheofone,...,2,1,1,,...,,

blockstodpartitioneis

:thatAssume

),(

),(),(,

,),,(

]0[1

]0[1

21

]1[11

]1[1

][1

]0[1

11

nn

nn

kp

n

innn

in

inn

nnnnnnnn

qq

ytJI

tJIy

qkqrqprrr

NBLOCKSy

ytftyyytJI

ytfftyyftyy

fyytfdt

dy

n y

3. Forming the partitioned system

]1[

11]1[

1][1

]0[1

]1[11

]1[1

][1

]0[1

][1

]0[

1

~]0[1

]0[1

]0[1

]0[1

]0[1

]0[1

]1[11

]1[1

][1

]0[1

,

,

),(

innn

in

inn

nnnnnn

n

nnnnn

nn

innn

in

inn

ztftzzzS

ytftyyyA

methodNewtonregularthebyobtained

solutionacceptedthebeyLet

StISwithWSA

tJIA

ytftyyyA

Splitting matrix A

A = S + WA = S + W

WSA

WW

WW

WW

S

S

S

SWW

WSW

WWS

0

0

0

00

00

00

3231

2321

1312

33

22

11

333231

232221

131211

4. Why partitioned system?

Advantages:Advantages:

1. It is not necessary to compute the non-zero 1. It is not necessary to compute the non-zero elements of the weak blockselements of the weak blocks

2. Several small matrices are to be factorized 2. Several small matrices are to be factorized instead of one big matrixinstead of one big matrix

3. Several small systems of linear algebraic 3. Several small systems of linear algebraic equations are to be solved at each Newton equations are to be solved at each Newton iteration (instead of one large system)iteration (instead of one large system)

5. Requirements

]1[

11]1[

1][1

]0[1

]1[11

]1[1

][1

]0[1

,

,

innn

in

inn

innn

in

inn

ztftzzzS

ytftyyyA

It is desirable to find the conditions under which the approximate solution of the second problem is closeto the approximate solution of the first one

largermuchnotis

sufficientnotis0z

smallis][1n

][1

][1

nn zy

Lemma 1

dy

zytfT

BWSD

STStB

SC

yDzyCzyBzy

innni

n

innn

in

ni

nnin

nn

ninnnn

inn

in

inn

1

0

]1[1

]1[1,1]1[

1

]1[1

]0[1

1]0[1

]1[1

]0[

1

~]1[

1

1]0[1

]1[1

1]0[11

][1

]1[11

]1[1

][1

]1[1

][1

][1

)1(

where

)(

methodnumericalsome

byfoundofionapproximat

),(ofsolutionexact

1][1

1111

nn

nnnnn

yy

ytftyyy

Lemma 2

][1

]1[1

1

1

][1

1

1

1

][1

]0[1

][1

1

][1

][1

][1

)(

nki

n

i

k

k

j

jin

nnn

i

k

k

j

jin

nn

i

j

jin

inn

yDB

zyCB

zyBzy

methodnumericalsome

byfoundofionapproximat

),(ofsolutionexact

1][1

1111

nn

nnnnn

yy

ytftyyy

“Local” error

][

10

1 maxj

nij

nBB

11 nB Major assumption

Theorem

][1

][1

][11

inn

inn

zy

zy

1. if the number of iterations is sufficiently large and2. if the error from the previous step is sufficiently small

“Global” error

)(,1

10,

max

max

10][

][

0

][

knkk

kk

ik

ik

EEB

CE

nkBB

k

k

k

Theorem

If 11][ EandB kk then the inequalities

][1

][1

][11

11 nnnnnn zyandzy

will be satisfied when sufficiently many iterations are performed

“Global” error - continuation

Theorem

E

EzyEzy

E

EzyEzy

EIf

nn

nn

nnv

nn

n

n

1

1

1

1

,1

1

001][

1][1

1

001][

11

1

1

nzyzy

nzyzy

EIf

n

n

nn

vnn

00][

1][1

00][

11

1

1

,1

Actual partitioning

1. Matrix 1. Matrix SS, which is obtained after the partitioning is a , which is obtained after the partitioning is a block-diagonal matrix containing block-diagonal matrix containing 2323 blocks. blocks.

2. The first diagonal block is a 2. The first diagonal block is a 13x1313x13 matrix. matrix.

3. The next 3. The next 2222 clocks are clocks are 1x11x1 matrices (a Newton iteration matrices (a Newton iteration procedure for scalar equations is used in this part).procedure for scalar equations is used in this part).

4. This partitioning was recommended by the chemists (based 4. This partitioning was recommended by the chemists (based on their knowledge of the chemical reactions)on their knowledge of the chemical reactions)

Variation of the key quantities for different scenarios

ScenarioScenario )( NB E

1 [2.4E-4, 3.3E-2] 1.03425 2 [3.7E-4, 2.1E-2] 1.02179 3 [5.8E-3, 5.7E-2] 1.06040 4 [1.3E-3, 4.3E-2] 1.04606 5 [4.3E-3, 2.5E-2] 1.02522 6 [5.6E-3, 6.4E-2] 1.06882

standN 301260,...,2,1

)(,)( 1111 nnnn BBBB

Variation of the key quantities for different scenarios

Variation of the key quantities for different stepsizes

StepsizeStepsize B E

30 2.5E-2 1.025 10 1.4E-2 1.014 1 2.6E-3 1.0026 0.1 3.6E-4 1.00036 0.01 5.3E-5 1.000053

5Scenario

Numerical Results

Method Method Computing timeComputing time

QSSA-1 12.85QSSA-1 12.85

QSSA-2 11.72QSSA-2 11.72

Euler Euler 15.1115.11

Trapez. 15.94Trapez. 15.94

RK-2 28.49RK-2 28.49

Part. dense 10.09 Based on EulerPart. dense 10.09 Based on EulerBetter accuracy with the classical numerical methodsBetter accuracy with the classical numerical methods

Accuracy results

Numer. Method Scenario 2 Scenario 6Numer. Method Scenario 2 Scenario 6

QSSA-1 3.20E-3 4.39E-1QSSA-1 3.20E-3 4.39E-1

QSSA-2 3.39E-3 3.86E-1QSSA-2 3.39E-3 3.86E-1

Euler 5.78E-4 3.57E-3Euler 5.78E-4 3.57E-3

Trapez 5.78E-4 3.57E-3Trapez 5.78E-4 3.57E-3

Part. dense 5.72E-4 3.26E-3Part. dense 5.72E-4 3.26E-3st 30 The chemical compound is ozone

Conclusions and open problems

1.We have shown that the 1.We have shown that the physicalphysical arguments for achieving arguments for achieving successful partitioning can be justified with clear algebraic successful partitioning can be justified with clear algebraic requirements.requirements.

2. There is a 2. There is a drawbackdrawback: if the chemical scheme is changed, : if the chemical scheme is changed, then the whole procedure has to be carried out for the new then the whole procedure has to be carried out for the new chemical scheme.chemical scheme.

3. 3. Automatic partitioningAutomatic partitioning is desirable.is desirable.