A Newton–Cotes quadrature approach for solving the aerosol coagulation equation

Atmospheric Environment 36 (2002) 583–589

Technical note

A Newton–Cotes quadrature approach for solving the aerosolcoagulation equation

Adrian Sandu

Department of Computer Science, Michigan Technological University, Houghton, MI 49931, USA

Received 5 May 2001; accepted 23 August 2001

Abstract

This paper presents a direct approach to solving the aerosol coagulation equation. Newton–Cotes formulas are usedto discretize the integral terms, and the semi-discrete system is built using collocation. A semi-implicit Gauss–Seideltime integration method is employed. The approach generalizes the semi-implicit method of Jacobson and is more

accurate. r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Aerosol dynamics; Coagulation; Newton–Cotes integration

1. Introduction

As our understanding expands, new processes areincorporated into air quality computer models. Oneexample is the particulate matter (aerosol) processes, the

importance of which is now widely recognized. Aerosolsare a priority focus area in environmental science due tothe leading role they play as a cause of adverse human

health, and their ability to scatter and absorb incomingsolar radiation and thus modify warming due togreenhouse gases and reduce visibility. To accurately

study the effects of aerosols it is necessary to resolveaerosol number and mass distributions as a function ofchemical composition and size. In this paper we developa numerical technique for solving the aerosol coagula-

tion equation. The method is of the stationary sectionaltype: the number densities are computed at a predefinedset of particle volumes (bin mean volumes). The

discretization is based on approximating the integralterms by Newton–Cotes sums, and imposing that theresulting equations hold exactly at the node points

(collocation). A bilinear system of coupled ordinarydifferential equations is obtained which can be advancedin time by the time-stepping algorithm of choice; here we

use the semi-implicit formula solved with Gauss–Seideliterations.

The paper is organized as follows: Section 2 gives anoverview of aerosol dynamic equations; Section 3presents the discretization technique while Section 4

shows numerical results obtained on a test problem withanalytical solution. Conclusions and future work arehighlighted in Section 5. Newton–Cotes integration is

reviewed in the Appendix.

2. Continuous particle coagulation equation

In this paper the continuous particle size distributionsare functions of particle volume (v) and time (t). For

simplicity we consider single component particles; thetechnique can be generalized to multiple components.The size distribution function (number density) of afamily of particles is denoted by nðv; tÞ; the number of

particles per unit volume of air with the volume betweenv and vþ dv is nðv; tÞ dv: Similar formulations can begiven in terms of mass and surface densities (Pilinis,

1990).The aerosol population undergoes a series of physical

transformations which change the number density

(Gelbard and Seinfeld, 1978a) One of the impor-tant physical processes is coagulation; the theoreticalE-mail address: [email protected] (A. Sandu).

1352-2310/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 1 3 5 2 - 2 3 1 0 ( 0 1 ) 0 0 4 4 4 - 7

coagulation equation for single-component particles is(Jacobson, 1999, Section 16)

@

@tnðv; tÞ ¼

1

2

Z v

0

bv�w;wnðv� w; tÞnðw; tÞ dw

� nðv; tÞZ

N

0

bv;wnðw; tÞ dw;

nðv; t ¼ 0Þ ¼ n0ðvÞ: ð1Þ

Here bv;w denotes the coagulation kernel; the first

integral term is of ‘‘Volterra’’ type and describes thecreation of particles of volume v by coagulation ofsmaller particles; the second term is of ‘‘Fredholm’’ typeand models the loss of volume v particles due to

coagulation. The equation is subject to a specified initialdistribution n0:

2.1. Previous work

2.1.1. Representations of the particle size distributionThree major approaches are used to represent the

size distribution of aerosols: continuous, discrete and

parameterized. In this paper we focus on continuousmodels; for computational purposes one needs to usefinite-dimensional approximations. In the sectional

approach the size domain vA½0;N� is divided intosize bins vA½V low

i ; Vhighi Þ: In each size bin j there

are nj particles per unit volume, all of them having

the same mean volume Vi: In the full-stationary structurethe number of particles in each bin njðtÞ is allowed tochange in time, but the particle volumes in each bin (Vj)

are not. Formally, the density function is n�ðv; tÞ ¼

Psj¼1 njðtÞ dðv� VjÞ; where dðvÞ is the Dirac delta

function.1

2.1.2. Coagulation equation

The integro–differential coagulation equation is diffi-cult to solve accurately, due to the fact that the limit ofintegration of the first term depends on the variable v

and the integrands are quadratic (the first term is ofnonlinear Volterra type in the terminology of integralequations). The algorithms proposed in the literature forthe coagulation equation include semi-implicit solutions,

finite element method, orthogonal collocation over finiteelements, J-space transformations, analytical solutions,etc. A nice survey of several popular numerical methods

for particle dynamics equations is given in Zhang et al.(1999).The standard discrete version of the coagulation

equation uses a monomer size distribution (the volumeof the particles in bin each i is a multiple of the smallestvolume, Vi ¼ iV1; i ¼ 1; 2;y). The semi-implicit

scheme to solve the discrete coagulation equation (Turco

et al., 1979) is discussed by Jacobson in (1999, Section16). The differential equation is discretized in time using

backward Euler formula, and the quadratic terms nj�c �ðtÞncðtÞ are replaced by the ‘‘linearized’’ version nj�c �ðtÞncðt� DtÞ; where Dt is the numerical time step size.

The scheme can be adapted to general size distributions,and admits a volume-conserving formulation. A combi-nation of cubic splines (coagulation) and moving finiteelement techniques (growth part) was used by Tsang and

Hippe (1988). Meng et al. (1998) present a size-resolvedand chemically resolved model for aerosol dynamics in amass density formulation. Gelbard and Seinfeld (1978a,

1978b, 1980) solve the coupled dynamic equations usingorthogonal collocation over finite elements. Lushnikov(1976) uses generating functions to solve analytically

the coagulation equation for particles consisting ofmonomers of two kinds, under the assumptions of aconstant coagulation kernel b and particular initial

distributions.

3. A direct discretization of the coagulation equation

The coagulation eq. (1) is in practice restricted toparticles having volumes in a finite range,

0oVminpvpVmaxoN:

@

@tnðv; tÞ ¼

1

2

Z v�Vmin

Vmin

bv�w;w nðv� w; tÞnðw; tÞ dw

� nðv; tÞZ Vmax

Vmin

bv;wnðw; tÞ dw: ð2Þ

Particle volumes are distributed over several orders ofmagnitude, and to better capture this logarithmic

volume coordinates are frequently employed

@

@tnðlog v; tÞ ¼

1

2

Z log ðv�VminÞ

log Vmin

bv�w;wnðlog ðv� wÞ; tÞ

� nðlogw; tÞw dðlog wÞ � nðlog v; tÞZ log Vmax

log Vmin

bv;wnðlog w; tÞw dðlog wÞ: ð3Þ

Consider s size bins of volumes Vmin ¼V1oV2o?oVs ¼ Vmax: The direct approach for sol-ving Eq. (2) is based on discretizing the integrals by a

quadrature formula with nodes V1;y;Vs

Z b

a

f ðvÞ dvEXsi¼1

x½a; b�i f ðViÞ: ð4Þ

The bin volumes (the nodes Vi) are given, but we need tospecify the appropriate weights xi:

1Recall that dðxÞ ¼ 0 for xa0; dð0Þ ¼ N; andRVjþeVj�e f ðxÞ

dðx� VjÞ dx ¼ f ðVjÞ:

A. Sandu / Atmospheric Environment 36 (2002) 583–589584

Replacing the integrals in Eq. (2) by Eq. (4) gives

@

@tnðv; tÞ ¼

1

2

Xj

x½V1 ;v�V1�j bv�Vj ;Vj nðv� Vj ; tÞnðVj ; tÞ

� nðv; tÞXj

z½V1 ;Vs�j bv;Vj nðVj ; tÞ; ð5Þ

where we allow different quadrature weights (x; z) ondifferent intervals. For example, if b is symmetric

(bv;w ¼ bw;v for all v;w) then the integrand bv�w;w n�ðv� w; tÞnðw; tÞ is periodic on ½Vmin; v� Vmin�: Conse-quently, the first term in Eq. (2) is the integral of aperiodic function on a period, and choosing xj the

trapezoidal rule weights on V1; v� V1] should provide avery accurate discretization.We now impose that Eq. (5) holds exactly at the node

points (collocation). This gives

@

@tnðVi; tÞ ¼

1

2

Xij¼1

x½V1 ;Vi�V1�j bVi�Vj ;Vj nðVi � Vj ; tÞnðVj ; tÞ

� nðVi; tÞXsj¼1

z½V1 ;Vs �j bVi ;Vj nðVj ; tÞ; 1pips:

ð6Þ

The quadrature weights xj ; zj will be determined in

order to achieve the desired order of accuracy.For a discrete coagulation equation we must express

(6) only in terms of number density at the node points.Eq. (6) considers particles of volume Vi � Vj (which

produce Vi when struck by a particle of volume Vj); ingeneral Vi � Vj falls in between two node point volumes

VcpVi � VjoVcþ1; c ¼ cði; jÞ

with c depending on i and j: Therefore nðVi � Vj ; tÞ mustbe approximated by polynomial interpolation (of thesame order as the underlying Newton–Cotes formula).For example, if the evaluation of the first integral uses

trapezoidal rule, nðVi � Vj ; tÞ can be computed by linearinterpolation without losing the approximation order,

acði; jÞ ¼Vcði; jÞþ1 � Vi þ VjVcði; jÞþ1 � Vcði; jÞ

;

nðVi � Vj ; tÞ ¼ acði; jÞncði; jÞðtÞ þ ð1� acði; jÞÞncði; jÞþ1ðtÞ:

With this the semi-discrete coagulation equation be-comes a system of s coupled, bilinear ordinary differ-ential equations:

@niðtÞ@t

¼1

2

Xij¼1

xijbVi�Vj ;Vj ðacði; jÞncði; jÞðtÞ

þ ð1� acði; jÞÞncði; jÞþ1ðtÞÞnjðtÞ

� niðtÞXsj¼1

zjbVi ;Vj njðtÞ; 1pips: ð7Þ

Here xij denotes the jth trapezoidal weight (node Vj) for

the interval ½V1;Vi � V1�; and zj the jth Newton–Cotesweight for the interval ½V1;Vs�:

3.1. The semi-implicit method

The semi-implicit method of Jacobson (1999, Section16:2) can be viewed as a particular Newton–Cotesapproach, using the rectangular integration formula. Let

NðVi; tÞ denote the total number of particles in size bin i;i.e. particles having volumes between Vi�1 and Vi: N�ðVi; tÞ are related to the density nðv; tÞ by

NðV1; tÞ ¼ V1nðV1; tÞ;

NðVi; tÞ ¼ ðVi � Vi�1ÞnðVi; tÞ; 2pips:

The solution is advanced from the time moment tk totkþ1 ¼ tk þ Dt using the following method:

Do i ¼ 1 : s

NðVi; tkþ1Þ ¼

NðVi; tkÞ þDt2

Pi�1j¼1

bVi�Vj ;Vj NðVi � Vj ; tkþ1ÞNðVj ; tkÞ

1þ DtPsj¼1

bVi ;Vj NðVj ; tkÞ:

ð8Þ

NðVi � Vj ; tkþ1Þ is obtained from fNðVm; tÞg1pmps bylinear interpolation.

3.2. Time integration

The semi-discrete system (7) could be integrated intime with an explicit or with an implicit time-stepping

formula. For implementing an implicit formula theJacobian of the derivative function is needed; theJacobian is easy to derive analytically, due to the

bilinear function form; we do not give more details here.Of interest is the direct generalization of the semi-

implicit method of Jacobson; the time-stepping idea can

be extended to a full Gauss–Seidel approach, which (inthe context of Newton–Cotes formulation) reads

Do i ¼ 1 : s

nðVi; tkþ1Þ

¼

nðVi; tkÞ þ Dt2

Pi�1j¼1

x½V1 ;Vi �j bVi�Vj ;Vj nðVi � Vj ; tkþ1Þ

nðVj ; tkþ1Þ þ Dt2 x

½V1 ;Vi �i b0;Vi nð0; t

kþ1ÞnðVi; tkÞ

1þ DtPi�1j¼1

z½V1 ;Vs �j bVi ;Vj nðVj ; t

kþ1Þ

þDtPsj¼i

z½V1 ;Vs�j bVi ;Vj nðVj ; t

kÞ

ð9Þ

Note that in real problems the density of zero-volumeparticles nð0; tkþ1Þ (which appears above) is zero due to

physical considerations; this term however is not zero inour test problem (see Section 4).

A. Sandu / Atmospheric Environment 36 (2002) 583–589 585

4. Numerical results

4.1. Test problem

For the numerical experiments we consider the test

problem from Gelbard and Seinfeld (1978a). Let Ntot bethe total initial number of particles and Vmean the meaninitial volume. The initial number distribution isexponential, and the coagulation rate is constant:

n0ðvÞ ¼ ðNtot=VmeanÞ e�v=Vmean ; bðv;wÞ ¼ b0:

This test problem admits the analytical solution(Gelbard and Seinfeld, 1978b)

nðv; tÞ ¼4Ntot

VmeanðNtotb0tþ 2Þ2exp

�2vVmeanðNtotb0tþ 2Þ

� �:

We solve this coagulation equation for Vmean ¼0:05 mm3; Ntot ¼ 104 particles=cm3; and

b0 ¼8kBT

3Za

��T¼298 K

¼ 6:017� 10�10 cm3=s particle:

(Za is the atmospheric dynamic viscosity). The integra-tion time interval is 24 h; and the integration time step

Dt ¼ 0:2 s is chosen small enough to ensure that timediscretization errors are negligible when compared tovolume discretization errors. For discretization of the

integral terms we choose the fourth order repeatedNewton–Cotes algorithm (Boole formula) as presented

in the Appendix.The initial and final number distributions are shown

in Fig. 1.

Experiment 1. In the first experiment we consider

particles with diameters between Dmin ¼ 0 mm andDmax ¼ 1 mm (i.e. volumes in the range ½Vmin ¼0 mm3;Vmax ¼ 5:236E � 1 mm3�; and use equidistant sizebin volumes,

Dv ¼Vmax � Vmin

s� 1; Vi ¼ Vmin þ ði � 1ÞDv; 1pips:

The computed solutions and the error distribution(versus particle diameter) at the end of the 24 h interval

are shown in Fig. 2. The fourth order Newton–Cotesdiscretization (Boole formula) provides more accuratesolutions than the semi-implicit method for the same

number of discretization points. Note that using thetrapezoidal discretization for the first integral term (witha linear interpolation for nðv� w; tÞ) gives better resultsthan the Boole formula with order four polynomialinterpolation (for reasons of periodicity, as explainedbefore). Increasing the number of bins beyond s ¼ 20does not seem to result in more accurate solutions; the

reason is that the upper integration limit in the

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

Particle Diameter [ µm ]

n [ (

part

icle

s/cm

3 ) /µ

m3 ]

103

102

101

100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105


n [ (

part

icle

s/cm

3 ) /µ

m3 ]

Initial, T = 0 hrsFinal, T = 24 hrs

- - -

Fig. 1. Exact initial and final distributions, in linear (left) and logarithmic (right) coordinates.


coagulation negative term is set to Vmax instead of N;and this approximation error becomes larger than thenumerical error itself.

Experiment 2. In the second experiment we considerparticles with diameters in between Dmin ¼ 1E � 3 mmand Dmax ¼ 1 mm (i.e. volumes in the range ½Vmin ¼5:236E � 10 mm3; Vmax ¼ 5:236E � 1 mm3�: The distri-bution of the bin volumes is logarithmic-uniform,

r ¼Vmax

Vmin

� �1=ðs�1Þ

; Vi ¼ Vminri�1; 1pips:

We actually solve form (3) of the coagulation equation,

with the interpolation also performed on logarithmicscale. Linear interpolation works better than higherorder interpolation.The computed solutions and the error distribution

(versus particle diameter) at the end of the 24 h intervalare shown in Fig. 3. Boole formula again provides moreaccuracy, but the difference is less dramatic. For a large

number of size bins the semi-implicit method is slightlybetter toward the high-volume end of the interval. Thisis probably due to the fact that the log-uniform grid has

insufficient resolution at high-particle volumes (dia-meters).

5. Conclusions and future work

In this paper, we developed a direct method for thesize discretion of aerosol coagulation equation. The

number densities are computed at a predefined set ofparticle volumes (bin mean volumes). The coagulationrate integral terms are replaced by discrete approxima-

tions provided by Newton–Cotes formulas. The result-ing equations are imposed to hold exactly at the nodepoints (collocation). This semi-discretization in sizeresults in a bilinear system of coupled ordinary differ-

ential equations. The system can be advanced in time bythe time-stepping method of choice.The proposed discretization methods are inexpensive,

since the weights can be calculated before the integrationprocess and stored. For implicit time-stepping theJacobian can be easily obtained; however it does not

have any special structure (it is a full matrix) so solvingthe linear systems might prove costly for large numberof bins. An extension of the semi-implicit time stepping

method to a full Gauss–Seidel approach was used in theexperiments for time integration.Future work will focus on improving the accuracy of

the quadrature discretizations for the logarithmic dis-

tribution of bin volumes.

0 0.61 0.79 0.91 10

5

10

15x 10

4


n [ (

part

icle

s/cm

3 ) /µ

m3 ] Exact

Boole, s=10Boole, s=20Boole, s=40

0 0.61 0.79 0.91 110

5

100


Rel

ativ

e E

rror


0 0.61 0.79 0.91 10

5

10

15x 10

4


n [ (

part

icle

s/cm

3 ) /µ

m3 ] Exact

JSI, s=10JSI, s=20JSI, s=40

0 0.61 0.79 0.91 110

5

100


Rel

ativ

e E

rror


- -

Fig. 2. Distribution of solution errors in Experiment 1 after 24 h: Compared are Jacobson’s original method, and Boole discretization.


Acknowledgements

This work was supported by NSF CAREER awardACI-0093139.

Appendix A. Newton–Cotes integration

A Newton–Cotes formula of order n for evaluatingR ba f ðvÞ dv is based on a repeated application of the follo-wing elementary rule. The order n polynomial pnðvÞ thatinterpolates the function at nodes v1 through vnþ1 isexpressed as pnðvÞ ¼

Pnþ1i¼1 f ðviÞciðvÞ; where ci are the

degree n Lagrange interpolation polynomials, ciðvjÞ ¼dij : ThenZ vnþ1

v1

f ðvÞ dvEZ vnþ1

v1

pnðvÞ dv ¼Xnþ1i¼1

f ðviÞZ vnþ1

v1

ciðvÞ dv

¼Xnþ1i¼1

wi f ðviÞ:

To cover the whole interval ½a; b� the elementary rule is

repeatedly applied on subintervals ½v1; vnþ1�; ½vnþ2; v2nþ2�etc. If the number of nodes is not a multiple of nþ 1 the

integration of the last interpolation polynomial isrestricted to the remaining subinterval. For convenience,

we present the weights for orders 1, 2 and 4.Trapezoidal rule ðn ¼ 1Þ:

w1 ¼ w2 ¼v2 � v1

2:

Simpson rule ðn ¼ 2Þ:

w1 ¼ðv3 � v1Þð2v1 � 3v2 þ v3Þ

6ðv1 � v2Þ; w2 ¼

ðv3 � v1Þ3

6ðv1 � v2Þðv2 � v3Þ;

w3 ¼ðv3 � v1Þðv1 � 3v2 þ 2v3Þ

6ð�v2 þ v3Þ:

Boole rule ðn ¼ 4Þ:

w1 ¼ ½ðv5 � v1Þð12v31 � 15v21v2 � 15v21v3 þ 20v1v2v3

� 15v21v4 þ 20v1v2v4 þ 20v1v3v4 � 30v2v3v4

þ 9v21v5 � 10v1v2v5 � 10v1v3v5 þ 10v2v3v5

� 10v1v4v5 þ 10v2v4v5 þ 10v3v4v5 þ 6v1v25

� 5v2v25 � 5v3v

25 � 5v4v

25 þ 3v35Þ�=½60ðv1 � v2Þ

� ðv1 � v3Þðv1 � v4Þ�;

103

102

101

100

0

5

10

15

x 104


n [ (

part

icle

s/cm

3 ) /µ

m3 ]

ExactBoole, s=10Boole, s=20Boole, s=40

103

102

101

100

102

101

100


Rel

ativ

e E

rror


103

102

101

100

0

5

10

15

x 104


n [ (

part

icle

s/cm

3 ) /µ

m3 ]

ExactJSI, s=10JSI, s=20JSI, s=40

103

102

101

100

102

101

100


Rel

ativ

e E

rror


- - - - - -

-

-- - -

-

-

- - -

Fig. 3. Distribution of solution errors in Experiment 2 after 24 h: Compared are Jacobson’s original method and Boole discretization.


w2 ¼

ðv5 � v1Þ3ð3v21 � 5v1v3 � 5v1v4 þ 10v3v4 þ 4v1v5

�5v3v5 � 5v4v5 þ 3v25Þ60ðv1 � v2Þðv2 � v3Þðv2 � v4Þðv2 � v5Þ

;

w3 ¼


�5v2v5 � 5v4v5 þ 3v25Þ60ðv1 � v3Þð�v2 þ v3Þðv3 � v4Þðv3 � v5Þ

;

w4 ¼


�5v2v5 � 5v3v5 þ 3v25Þ60ðv1 � v4Þð�v2 þ v4Þð�v3 þ v4Þðv4 � v5Þ

;

w5 ¼ ½ðv5 � v1Þð3v31 � 5v21v2 � 5v21v3 þ 10v1v2v3 � 5v21v4

þ 10v1v2v4 þ 10v1v3v4 � 30v2v3v4 þ 6v21v5 � 10v1v2v5

� 10v1v3v5 þ 20v2v3v5 � 10v1v4v5 þ 20v2v4v5

þ 20v3v4v5 þ 9v1v25 � 15v2v

25 � 15v3v

25 � 15v4v

25 þ 12v35Þ�=

½60ð�v2 þ v5Þð�v3 þ v5Þð�v4 þ v5Þ�:

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A Newton–Cotes quadrature approach for solving the aerosol coagulation equation