Convergence analysis of sectional methods for solving ...population balance equations: The ﬁxed...

Convergence analysis of sectional methods for solving aggregation

population balance equations: The fixed pivot technique

Ankik Kumar Giri∗ , Jitendra Kumar and Gerald Warnecke

Institute for Analysis and Numerics, Otto-von-Guericke University Magdeburg,

Universitätsplatz 2, D-39106 Magdeburg, Germany

November 16, 2009

Abstract. In this paper, we introduce the convergence analysis of the fixed pivot technique

given by S. Kumar and Ramkrishna [22] for the nonlinear aggregation population balance

equations which are of substantial interest in many areas of science: colloid chemistry, aerosol

physics, astrophysics, polymer science, oil recovery dynamics, and mathematical biology.

In particular, we investigate the convergence for five different types of uniform and non-

uniform meshes which turns out that the fixed pivot technique is second order convergent on

a uniform and non-uniform smooth meshes. Moreover, it yields first order convergence on a

locally uniform mesh. Finally the analysis exhibits that the method does not converge on an

oscillatory and non-uniform random meshes. The mathematical results of the convergence

analysis are also justified numerically.

Keywords: Particles; Aggregation; Fixed pivot technique; Consistency; Convergence.

Mathematics Subject Classification (2000) 45J05, 65R20, 45L10

1 Introduction

The continuous aggregation population balance equation (PBE) or Smoluchowski coagulationequation describes the kinetics of particle growth in which particles can aggregate via binaryinteraction to form larger particles. This model arises in many applications such as clusterformation in galaxies, kinetics of phase transformations in binary alloys, aggregation of red bloodcells, fluidized bed granulation processes etc. The nonlinear continuous aggregation populationbalance equation is given by, see e.g. [5]

∂f(t, x)

∂t=

1

2

∫ x

0β(x − ǫ, ǫ)f(t, x − ǫ)f(t, ǫ)dǫ −

∫ ∞

0β(x, ǫ)f(t, x)f(t, ǫ)dǫ, (1)

with

f(x, 0) = f in(x) ≥ 0, x ∈]0,∞[.∗Corresponding author. Tel +49 391 6711629; Fax +49 391 6718073

Email address: [email protected]

1

where the variables x > 0 and t ≥ 0 denote the size of the particles and time respectively. Thenumber density of particles of size x at time t is denoted by f(x, t) ≥ 0. The aggregation kernelβ(x, ǫ) ≥ 0 represents the rate at which particles of size x coalesce with those of size ǫ. It willbe assumed throughout that β(x, ǫ) = β(ǫ, x) for all x, ǫ > 0, i.e. symmetric.Mathematical results on existence and uniqueness of solutions of equation (1) can be found in[7, 10, 24, 25, 28, 29, 31, 33] for different classes of aggregation kernels. However, for the sakeof simplicity in our analysis we consider them to be twice continuously differentiable functions.The aggregation PBE (1) can be solved analytically only for some restricted class of aggregationkernels, see [6, 14, 15]. Because of restrictions, we need to solve it numerically. In the PBE (1)the volume variable x ranges from 0 to ∞. In order to apply a numerical scheme to the solutionof the equation, the initial step is to set a finite computational domain. In this work we considerthe following truncated equation

∂n(t, x)

∂t=

1

2

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫ −

∫ xmax

0β(x, ǫ)n(t, x)n(t, ǫ)dǫ, (2)

with

n(x, 0) = nin(x) ≥ 0, x ∈ Ω :=]0, xmax].

Here the variable n(t, x) represents the solution to the preceding truncated equation. Theexistence and uniqueness of non-negative solutions for the truncated PBE (2) has been derivedin [2, 4, 7, 33].Many numerical methods have been proposed to solve the truncated PBE (2): finite elementmethods [11, 30, 32], finite volume methods [3, 12, 13], stochastic methods [8, 9, 26], momentmethods [34] and sectional methods [17, 18, 22, 23].A large variety of finite element methods, finite volume methods, weighted residuals, the methodof orthogonal collocation and Galerkin’s method are implemented for solving aggregation PBEs.By using these methods, we may have a good prediction of number density but a poor predictionof moments, see J. Kumar et al. [21]. These methods are computationally very expensive andinclude stability problems. In moment methods, the PBE is transformed into a system of ODEsdescribing the evolution of the moments of the particle size distribution. The moment methodsindeed predict very accurately the moments of the distribution but are unable to give preciseinformation about the density distribution. In recent times, the sectional methods have becomecomputationally very attractive because they not only predict accurately some selected momentsof the distribution, but also give satisfactory results for the complete density distribution.Several authors have proposed sectional methods for aggregation PBE: S. Kumar and Ramkr-ishna [22, 23], J. Kumar et al. [17, 18] and Vanni [35]. Among all sectional methods, the fixedpivot technique is the most extensively used method these days due to its generality and robust-ness. This technique predicts the first two moments of the distribution very accurately. Despitethe fact that the first two numerically calculated moments are fairly accurate, the fixed pivottechnique consistently over-predicts the results of number density as well as its higher moments.A step to improve the fixed pivot technique by preserving all advantages of the existing sectionalmethods has been recently made by the J. Kumar et al. [17, 18] in the cell average technique.

2

xi−1/2 xi+1/2

xi−1 xi+1xi

(i − 1)th cell ith cell (i + 1)th cell

xi+3/2xi−3/2

∆xi

Figure 1: A discretized size domain.

The purpose of this work is to demonstrate the missing numerical analysis of sectional methodsfor aggregation PBEs in the literature. Recently J. Kumar and Warnecke [19, 20] have publishedthe numerical analysis of sectional methods for breakage PBEs. This case was simpler due to thelinearity of that equation. In this work we examine the method of S. Kumar and Ramkrishna[22] for the nonlinear case of aggregation of particles.Let us now briefly outline the contents of this paper. The general idea of sectional methods aredescribed in next section where some useful definitions and an existing theorem used in furtheranalysis of these methods are also stated. The mathematical formulation of the fixed pivottechnique is reviewed in Section 3. The consistency and convergence are discussed in Sections 4and 5, respectively. Numerical experiments are performed in Section 6. And, some conclusionsare formed in Section 7.

2 The Sectional Methods

Sectional methods can be described by the following general mathematical derivation. Thesemethods approximate the total number of particles in finite number of cells. As a first step, thecontinuous interval Ω :=]0, xmax] is divided into a small number of cells defining size classes

Λi :=]xi−1/2, xi+1/2], i = 1, . . . , I,

withx1/2 = 0, xI+1/2 = xmax, ∆xi = xi+1/2 − xi−1/2 ≤ ∆x.

The representative of each size class, usually the center point of each cell xi = (xi−1/2+xi+1/2)/2,is called pivot or grid point. This type of partitioning of the spatial domain is known as cellcentered representation of the mesh. A typical cell centered partitioning of the domain is shownin Figure 1. The integration of the equation (2) over each cell yields a semi-discrete system inR

I

dN

dt= B − D, (3)

N(0) = Nin.

Here we consider Nin,N,B,D ∈ RI whose semi-discrete ith components are defined as

Ni(t) =

∫ xi+1/2

xi−1/2

n(t, x)dx, (4)

3

N ini =

∫ xi+1/2

xi−1/2

nin(x)dx,

Bi =1

2

∫ xi+1/2

xi−1/2

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx. (5)

and

Di =

∫ xi+1/2

xi−1/2

∫ xI+1/2

0β(x, ǫ)n(t, ǫ)n(t, x)dǫdx. (6)

Note that N is the vector of values of the step function obtained by L2 projection of the exactsolution n into the space of step functions constant on each cell. Various sectional methods forthe numerical solutions of the equation (2) can be obtained from different choices of numericalapproximations of Bi and Di in terms of Ni(t). Finally the sectional methods take the followingspatially discretized form

dN̂

dt= B̂(N̂) − D̂(N̂) =: F̂(t, N̂), (7)

N̂(0) = Nin, (8)

where B̂, D̂ ∈ RI are some functions of N̂. The ith component, N̂i(t) of the vector N̂ is thenumerical approximation of the total number in ith cell Ni(t).

Before we proceed to the next section to discuss the fixed pivot technique in detail, let us reviewsome useful definitions and a theorem given in Hundsdorfer and Verwer [16].Let ‖ · ‖ denote any norm on RI .

Definition 2.1. The spatial truncation error is defined by the residual left by substitutingthe exact solution N(t) into equation (7) as

σ(t) =dN(t)

dt−(

B̂ (N(t)) − D̂ (N(t)))

. (9)

The scheme (7) is called consistent of order p if, for ∆x → 0,

‖σ(t)‖ = O(∆xp), uniformly for all t, 0 ≤ t ≤ T.

Definition 2.2. The global discretization error is defined by

ǫ(t) = N(t) − N̂(t). (10)

The scheme (7) is called convergent of order p if, for ∆x → 0,

‖ǫ(t)‖ = O(∆xp), uniformly for all t, 0 ≤ t ≤ T.

In the following theorem we consider M̂ ≥ 0 for a vector M̂ ∈ RI if all its components arenon-negative.

4

Theorem 2.3. (Hundsdorfer and Verwer [16]). Suppose that F̂(t,M̂) is continuous and satisfiesthe Lipschitz condition

‖F̂(t, P̂) − F̂(t,M̂)‖ ≤ L‖P̂ − M̂‖ for all P̂,M̂ ∈ RI .

Then the solution of the semi-discrete system (7) is non-negative if and only if for any vectorM̂ ∈ RI and all i = 1, . . . , I and t ≥ 0,

M̂ ≥ 0, M̂i = 0 =⇒ F̂i(t,M̂) ≥ 0.

Proof. The proof can be found in Hundsdorfer and Verwer [16], Chap. 1, Theorem 7.1.

Before proceeding to the next section it should be mentioned here that in this work we considerthe following L1 norm

‖N‖ =I∑

i=1

|Ni|.

3 The Fixed Pivot Technique

The fixed pivot technique is based on the idea of birth modification. According to Kumar andRamkrishna [22], the equation (2) is modified to

d

dt

∫ xi+1/2

xi−1/2

n(t, x)dx ≈12

∫ xi+1

xi

λ+i (x)

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx

+1

2

∫ xi

xi−1

λ−i (x)

∫ x


−∫ xi+1/2

xi−1/2

∫ xI+1/2

0β(x, ǫ)n(t, ǫ)n(t, x)dǫdx.

where

λ±i (x) =x − xi±1xi − xi±1

. (11)

Now sustituting the number density approximation

n(t, x) ≈I∑

i=1

Ni(t)δ(x − xi),

into the preceding equation, we obtain the following spatially discretized system

dNi(t)

dt=

j≥k∑

xi≤xj+xk

where the first two and last terms on the right hand side of (12) represent the birth and deathterms obtained from the fixed pivot technique respectively. The fundamental concept of thefixed pivot technique can be summarized as follows. Suppose that a new particle of a size,which is not a pivot point of any cell, appears due to the aggregation of two smaller particles.The particle has to be divided onto neighboring pivot points in such a way that the particlenumber and mass are conserved. This problem is uniquely solvable. The resulting techniquevery often gives quite satisfactory results. Moreover, we shall observe later that the fixed pivottechnique is unfortunately a zero order method on oscillatory and non-uniform random meshesfor aggregation problems.

4 Consistency

We need the following lemma to investigate the consistency of the fixed pivot technique.

Lemma 4.1. Consider a function f ∈ C2([0, xmax]) and a cell centered partitioning of thedomain [0, xmax] given as 0 = x1−1/2 < . . . < xi−1/2 < xi+1/2 < . . . < xI+1/2 = xmax with pivot

points xi = (xi−1/2 + xi+1/2)/2 and a bound ∆x ≥ ∆xi = (xi+1/2 − xi−1/2) for all i. If λ+i (x)and λ−i (x) are given by the relation (11), then the following approximations can be obtained

∫ xi+1/2

xi−1/2

f(x)dx =

∫ xi+1

xi

λ+i (x)f(x)dx +

∫ xi

xi−1

λ−i (x)f(x)dx

+f(xi)

2

[

∆xi −(

∆xi−1 + ∆xi+12

)]

− f′(xi)

12

[

(∆xi+1 − ∆xi−1){

∆xi +

(

∆xi−1 + ∆xi+12

)}]

+ O(∆x3), i = 2, . . . , I − 1,∫ xi+1/2

xi−1/2

f(x)dx =

∫ xi

xi−1

λ−i (x)f(x)dx +f(xi)

2[∆xi − ∆xi−1] + O(∆x2), i = I,

∫ xi+1/2

xi−1/2

f(x)dx =

∫ xi+1

xi

λ+i (x)f(x)dx +f(xi)

2[∆xi − ∆xi+1] + O(∆x2), i = 1.

Proof. The proof can be found in [19] by applying Taylor expansions to the following expressionsthat we introduce here in order to use this notation later

Ji(f) =

∫ xi+1/2

xi−1/2

f(x)dx −(∫ xi+1

xi

λ+i (x)f(x)dx +

∫ xi

xi−1

λ−i (x)f(x)dx

)

for i = 2, . . . , I − 1,

J1(f) =

∫ xi+1/2

xi−1/2

f(x)dx −∫ xi+1

xi

λ+i (x)f(x)dx for i = 1,

and

JI(f) =

∫ xi+1/2

xi−1/2

f(x)dx −∫ xi+1

xi

λ−i (x)f(x)dx for i = I.

6

Now we describe the three main subsections to inquire the consistency of the fixed pivot tech-nique. We evaluate the order of the birth term and the order of the death term with the briefsummary of all terms in Subsections 4.1 and 4.2, respectively . Finally, we consider the fivedifferent types of meshes to evaluate the local discretization error in Subsection 4.3.

4.1 Order of the birth term

The integrated birth term can be written as follows

Bi =1

2

∫ xi+1/2

xi−1/2

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx.

Let us denote

f(t, x) =1

2

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫ.

4.1.1 Birth term on internal cells

Considering i = 2, . . . , I − 1 and using Lemma 4.1, we can rewrite Bi as

Bi =1

2

∫ xi+1

xi

λ+i (x)

∫ x


+1

2

∫ xi

xi−1

λ−i (x)

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx + Ji(f).

By changing the order of integration of the first two terms on the right hand side, we obtain

Bi =1

2

∫ xi

0

∫ xi+1

xi

λ+i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ xi+1

xi

∫ xi+1

ǫλ+i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ xi−1

0

∫ xi

xi−1

λ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ xi

xi−1

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + Ji(f).

7

Each integral term on the right hand side can be further rewritten as

Bi =1

2

i−1∑

j=1

∫ xj+1/2

xj−1/2

∫ xi+1

xi


+1

2

∫ xi

xi−1/2

∫ xi+1

xi


+1

2

∫ xi+1

xi

∫ xi+1

ǫλ+i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

i−2∑

j=1

∫ xj+1/2

xj−1/2

∫ xi

xi−1


+1

2

∫ xi−1

xi−3/2

∫ xi

xi−1


+1

2

∫ xi

xi−1

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + Ji(f). (13)

Let us denote the integral terms on the right hand side by I1, . . . , I6 respectively and calculatethem separately.

The integrals I6 and I3.

I6 =1

2

∫ xi

xi−1

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

=1

2

∫ xi

xi−1

g(t, ǫ)dǫ,

where

g(t, ǫ) :=

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dx. (14)

Using Taylor series expansion of g(t, ǫ) with respect to ǫ about xi−1, we get

I6 = g(t, xi−1)(∆xi−1 + ∆xi)

4+ gǫ(t, xi−1)

(∆xi−1 + ∆xi)2

16+ . . . , (15)

where

gǫ(t, xi−1) =∂g

∂ǫ(t, xi−1) =

∂g

∂ǫ(t, ǫ)|ǫ=xi−1 .

Now we calculate gǫ(t, xi−1) using the Leibniz rule to differentiate under an integral using

8

λ−i (xi−1) = 0

gǫ(t, xi−1) =

[

nǫ(t, ǫ)

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)dx

+ n(t, ǫ)∂

∂ǫ

∫ xi

ǫλ−i (x)β(x − ǫ, ǫ)n(t, x − ǫ)dx

]

ǫ=xi−1

,

= nǫ(t, xi−1)

∫ xi

xi−1

λ−i (x)β(x − xi−1, xi−1)n(t, x − xi−1)dx

+n(t, xi−1)

[∫ xi

ǫλ−i (x)

∂

∂ǫ{β(x − ǫ, ǫ)n(t, x − ǫ)}dx

]

ǫ=xi−1

,

= nǫ(t, xi−1)

∫ xi

xi−1

λ−i (x)β(x − xi−1, xi−1)n(t, x − xi−1)dx

+n(t, xi−1)

∫ xi

xi−1

λ−i (x)

[

{ ∂∂ǫ

β(x − ǫ, ǫ)}ǫ=xi−1n(t, x − xi−1)

+ β(x − xi−1, xi−1)nǫ(t, x − xi−1)]

dx.

Then, we apply the left rectangle rule to get

gǫ(t, xi−1) = 0 + 0 + O(∆x2).

Now we have to evaluate g(t, xi−1) from the equation (14)

g(t, xi−1) =

∫ xi

xi−1

λ−i (x)β(x − xi−1, xi−1)n(t, x − xi−1)n(t, xi−1)dx.

Again apply the left rectangle rule, and we obtain

g(t, xi−1) = 0 + O(∆x2).

By substituting the value of gǫ(t, xi−1) and g(t, xi−1) in the equation (15) we get

I6 = 0 + O(∆x3). (16)

Similarly, we can evaluate the third term

I3 = 0 + O(∆x3). (17)

The integrals I1 and I4.Now we consider the first term in (13)

I1 =1

2

i−1∑

j=1

∫ xj+1/2

xj−1/2

∫ xi+1

xi


9

and apply the midpoint rule to get

I1 =1

2

i−1∑

j=1

∫ xi+1

xi

λ+i (x)β(x − xj, xj)n(t, x − xj)dx · n(t, xj)∆xj + O(∆x3).

Furthermore we can use the relationship Ni = n(t, xi)∆xi + O(∆x3) for the mid point rule toget the form

I1 =1

2

i−1∑

j=1

Nj

∫ xi+1

xi

λ+i (x)β(x − xj , xj)n(t, x − xj)dx + O(∆x3).

By using the substitution x − xj = x′ we obtain

I1 =1

2

i−1∑

j=1

Nj

∫ xi+1−xj

xi−xj

λ+i (x′ + xj)β(x

′, xj)n(t, x′)dx′ + O(∆x3). (18)

We define li,j and γi,j to be those indicies such that the following hold

xi − xj ∈ Λli,j and γi,j := H[(xi − xj) − xli,j ] (19)

where

H(x) :=

{

1 if x > 0,

-1 if x ≤ 0.

We will use the convention of Riemann integration that∫ ba f(x)dx = −

∫ ab f(x)dx and the

equation (18) can be rewritten as

I1 =1

2

i−1∑

j=1

Nj

∫ xli,j+

12

γi,j

xi−xj


′, xj)n(t, x′)dx′

+1

2

i−1∑

j=1

Nj

li+1,j+1

2(γi+1,j−1)∑

k=li,j+1

2(1+γi,j )

∫ xk+1/2

xk−1/2



+1

2

i−1∑

j=1

Nj

∫ xi+1−xj

xli+1,j+

12

γi+1,j


′, xj)n(t, x′)dx′ + O(∆x3).

We apply the midpoint rule in the second term on the right hand side, and get

I1 =1

2

i−1∑

j=1

Nj

∫ xli,j+

12

γi,j

xi−xj



+1

2

i−1∑

j=1

Nj

li+1,j+1

2(γi+1,j−1)∑

k=li,j+1

2(1+γi,j )

λ+i (xk + xj)β(xk, xj)n(t, xk)∆xk

+1

2

i−1∑

j=1

Nj

∫ xi+1−xj

xli+1,j+

12

γi+1,j


′, xj)n(t, x′)dx′ + O(∆x3).

10

This can be further rewritten as

I1 =1

2

i−1∑

j=1

Nj

∫ xli,j+

12

γi,j

xi−xj



+1

2

i−1∑

j=1

Nj∑

xi≤xj+xk

The integral term on the right hand side can be rewritten as

I2 =1

2

∫ xi+1

xi

λ+i (x)β(x − xi, xi)n(t, x − xi)Nidx

− 14

∫ xi+1

xi

λ+i (x)β(x − xi, xi)n(t, x − xi)Nidx + O(∆x3).

When we apply the right rectangle rule in the second integral on the right hand side usingλ+i (xi+1) = 0, we obtain

I2 =1

2

∫ xi+1

xi

λ+i (x)β(x − xi, xi)n(t, x − xi)Nidx + O(∆x3).

Using the substitution x − xi = x′, we get

I2 =1

2Ni

∫ xi+1−xi

0λ+i (x

′ + xi)β(x′, xi)n(t, x

′)dx′ + O(∆x3),

=1

2Ni

li+1,i+1

2(γi+1,i−1)∑

k=1

∫ xk+1/2

xk−1/2

λ+i (x′ + xi)β(x

′, xi)n(t, x′)dx′

+1

2Ni

∫ xi+1−xi

xli+1,i+

12

γi+1,i


′, xi)n(t, x′)dx′ + O(∆x3).

Using the midpoint rule in the first term on the right hand side, we obtain

I2 =1

2Ni

li+1,i+1

2(γi+1,i−1)∑

k=1

λ+i (xk + xi)β(xk, xi)n(t, xk)∆xk

+1

2Ni

∫ xi+1−xi

xli+1,i+

12

γi+1,i


′, xi)n(t, x′)dx′ + O(∆x3).


I2 =1

2Ni

∑

xi+xk

The integral terms on the right hand side can be rewritten as

I5 =1

2Ni−1

∑

xi−1+xk

By combining some terms this can be further rewritten as

Bi =

j≥k∑

xi≤xj+xk

There may be one of the following possibilities:(1) xi − xj lies in the right half of the lthi,j cell i.e.

xli,j < xi − xj ≤ xli,j+1/2

or(2) xi − xj lies in the left half of the li,j th cell i.e.

xli,j−1/2 < xi − xj ≤ xli,j .

If xli,j < xi − xj ≤ xli,j+1/2 then from the equation (25) and using (xi+1 − xj) − (xi − xj) =12(∆xi + ∆xi+1), we obtain

|E1| ≤1

2

i−1∑

j=1

Nj

∫ (xi−xj)+(∆xi+∆xi+1)

xi−xj


′, xj)n(t, x′)dx′.

By applying the midpoint rule and using λ+i (xi+1) = 0, we get

|E1| ≤ 0 + O(∆x3).

If xli,j−1/2 < xi − xj ≤ xli,j then

|E1| =1

2

i−1∑

j=1

Nj

∫ xi−xj

xli,j−1/2



≤ 12

i−1∑

j=1

Nj

∫ xli,j−1/2+2∆+(∆xi+∆xi+1)

xli,j−1/2



where ∆ = (xi − xj) − xli,j−1/2. Again by applying the mid point rule and using λ+i (xi+1) = 0,we obtain

|E1| ≤ 0 + O(∆x3).

Similarly as before in E1 and using λ−i (xi−1) = 0, we can get

|E3| ≤ 0 + O(∆x3).

The integrals E2 and E4.Now we consider

E2 =1

2

i∑

j=1

Nj

∫ xi+1−xj

xli+1,j+

12

γi+1,j


′, xj)n(t, x′)dx′.

By the definition of the indicies li,j in (19)we can conclude that

xli+1,j−1/2 < xi+1 − xj ≤ xli+1,j+1/2.

15

There may be one of the following possibilities:(1) xi+1 − xj lies in the right half of the lthi+1,j cell i.e.

xli+1,j < xi+1 − xj ≤ xli+1,j+1/2

or(2) xi+1 − xj lies in the left half of the li+1,j th cell i.e.

xli+1,j−1/2 < xi+1 − xj ≤ xli+1,j .

If xli+1,j < xi+1 − xj ≤ xli+1,j+1/2 then

|E2| =1

2

i∑

j=1

Nj

∫ xli+1,j+1/2

xi+1−xj



≤ 12

i∑

j=1

Nj

∫ xli+1,j+1/2

xli+1,j+1/2−2∆1



where ∆1 = xli+1,j+1/2 − (xi+1 − xj). By applying the midpoint rule and using λ+i (xi+1) = 0,we obtain

|E2| ≤ 0 + O(∆x3).

If xli+1,j−1/2 < xi+1 − xj ≤ xli+1,j then

|E2| ≤1

2

i∑

j=1

Nj

∫ xli+1,j−1/2+2∆2

xli+1,j−1/2



where ∆2 = (xi+1 − xj)− xli+1,j−1/2.By applying the midpoint rule and using λ+i (xi+1) = 0, weobtain

|E2| ≤ 0 + O(∆x3).

Similarly as before in E2 and using λ−i (xi−1) = 0, we can get

|E4| ≤ 0 + O(∆x3).

Finally all these values can be substituted in equation (24)

Bi = BFPi + O(∆x3) + Ji(f).

After substituting the value of Ji(f) we get for i = 2, . . . , I − 1

Bi =BFPi +

f(xi)

2

[

∆xi −(

∆xi−1 + ∆xi+12

)]

− f′(xi)

12

[

(∆xi+1 − ∆xi−1){

∆xi +

(

∆xi−1 + ∆xi+12

)}]

+ O(∆x3). (26)

16

4.1.2 Birth term on boundary cells

Now we consider the birth term for i = 1

B1 =1

2

∫ x1+1/2

x1−1/2

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx.

Using the Lemma 4.1, the birth term can be rewritten as

B1 =1

2

∫ x2

x1

λ+1 (x)

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx + J1(f).

Changing the order of integration we get

B1 =1

2

∫ x1

0

∫ x2

x1

λ+1 (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ x2

x1

∫ x2

ǫλ+1 (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + J1(f).


B1 =1

2

∫ x1/2

0

∫ x2

x1


+1

2

∫ x1

x1/2

∫ x2

x1


+1

2

∫ x1+1/2

x1

∫ x2

ǫλ+1 (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ x2

x1+1/2

∫ x2

ǫλ+1 (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + J1(f).

We apply the right rectangle rule in the inner integral of the first and second term as well as leftand right rectangle rules in the outer integral of the third and fourth terms respectively to get

B1 =1

4∆x1

∫ x2

x1

λ+1 (x)β(x − x1, x1)n(t, x − x1)n(t, x1)dx + O(∆x2) + J1(f).

Again by applying the right rectangle rule we obtain

B1 =1

2N1

∑

x1+xk

Using the Lemma 4.1, the birth term can be rewritten as

BI =1

2

∫ xI

xI−1

λ−I (x)

∫ x

0β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dǫdx + JI(f).

After changing the order of integration, we get

BI =1

2

∫ xI−1

0

∫ xI

xI−1

λ−I (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ xI

xI−1

∫ xI

ǫλ−I (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + JI(f).

We break the first and the second integral into two parts as

BI =1

2

I−2∑

k=1

∫ xk+1/2

xk−1/2

∫ xI

xI−1


+1

2

∫ xI−1

xI−3/2

∫ xI

xI−1


+1

2

∫ xI−1/2

xI−1

∫ xI

ǫλ−I (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ

+1

2

∫ xI

xI−1/2

∫ xI

ǫλ−I (x)β(x − ǫ, ǫ)n(t, x − ǫ)n(t, ǫ)dxdǫ + JI(f).

By applying the midpoint, right, left and right rectangle rules to the outer integral of the first,second, third and fourth term respectively, we obtain

BI =1

2

I−2∑

k=1

∆xk

∫ xI

xI−1

λ−I (x)β(x − xk, xk)n(t, x − xk)n(t, xk)dx + 0

+1

4∆xI−1

∫ xI

xI−1

λ−I (x)β(x − xI−1, xI−1)n(t, x − xI−1)n(t, xI−1)dx + 0 + O(∆x2) + JI(f).


BI =1

2

I−2∑

k=1

Nk

∫ xI

xI−1

λ−I (x)β(x − xk, xk)n(t, x − xk)dx

+1

2NI−1

∫ xI

xI−1

λ−I (x)β(x − xI−1, xI−1)n(t, x − xI−1)dx + O(∆x2) + JI(f).

The first and second term on the right hand side can be solved similar to (I4) and (I5) respec-tively. Thus, we obtain

BI =1

2

I−2∑

j=1

Nj∑

xI−1≤xj+xk

i.e.,

BI =

j≥k∑

xI−1≤xj+xk

ξ

x

smooth transformation

x = g(ξ)

uniform mesh

non-uniform mesh

for examplex = exp(ξ)

Figure 2: Non-uniform smooth mesh.

x

uniform mesh uniform mesh

uniform mesh uniform mesh

�

I1

I2

�

I5

I6

]

I4

~

I3

~

I7

Figure 3: Locally uniform smooth mesh.

4.3 Meshes

Now let us consider the following five different types of meshes to calculate the order of localdiscretization error. Details can be found in J. Kumar and Warnecke [19].

4.3.1 Uniform mesh

Let us first consider uniform mesh i.e. ∆xi = ∆x. From the equations (30-32) we have ‖σ(t)‖ =O(∆x2). Thus the method is second order consistent on a uniform mesh.

4.3.2 Non-uniform smooth mesh

If we assume grids to be smooth in the sense that ∆xi −∆xi−1 = O(∆x2) and 2∆xi − (∆xi−1 +∆xi+1) = O(∆x3), where ∆x is the maximum mesh width, then similarly to the uniform case weagain have second order accuracy. Such grids can be obtained by some smooth transformationfrom uniform grids. Let us consider a variable ξ with uniform grids and a smooth transformationx = g(ξ) to get non-uniform smooth mesh, see Figure 2. The equations (30-32) can be simplifiedby using Taylor series expansion in the smooth transformation, see J. Kumar and Warnecke [19].So analogously to the uniform mesh we obtain ‖σ(t)‖ = O(∆x2), i.e. the technique is secondorder consistent.

4.3.3 Locally uniform mesh

An example of a locally uniform mesh is considered in Figure 3. Let us consider that thecomputational domain is divided into finitely many sub-domains and in each sub-domain isdivided into an equal size mesh. In this way we have a locally uniform mesh. In each cellexcept boundaries of every sub-domain we have third order accuracy due to uniform mesh ineach sub-domain. At all boundary cells of the sub-domains except the outer boundaries of the

20

first and last cell we have first order accuracy. This is due to the abrupt change in size at thisboundary. At the first (last) outer boundary we obtain a second order accuracy as in the caseof a uniform as well as non-uniform smooth mesh. For the example shown in the Figure 3 wehave ‖σ(t)‖ = O(∆x). Thus in this case we get only first order consistency.

4.3.4 Oscillatory mesh

Let us now consider oscillatory mesh i.e.

∆xi+1 :=

{

2∆xi if i is odd,12∆xi if i is even.

From the equations (30-32) we have

σi(t) = O(∆x), i = 1, . . . , I, and ‖σ(t)‖ = O(1).

Thus the fixed pivot method is unfortunately inconsistent on oscillatory meshes. This type ofmesh was not considered in J. Kumar and Warnecke [19] and [20] for breakage problems. Wehave repeated such meshes on the local discretization error obtained in the above mentionedpapers. We observed that the fixed pivot method is inconsistent on such meshes and the cellaverage technique is of first order only. These observations have been also verified numerically.

4.3.5 Non-uniform random mesh

Finally we analyze the scheme for random grids. Since no cancelation takes place in the leadingerror terms of equations (30-32) we have ‖σ(t)‖ = O(1). Thus the method is inconsistent onnon-uniform random meshes.

5 Convergence

First of all we shall prove the Lipschitz condition on B̂(N(t)) and D̂(N(t)) as follows:Let us consider the birth term for 0 ≤ t ≤ T and for all N, Ñ ∈ RI

‖B̂(N) − B̂(Ñ)‖ =I∑

i=1

|B̂i(N) − B̂i(Ñ)|.

Using βj,k ≤ C, due to the continuity of β and the finiteness of the domain, and 0 ≤ λ±i (x) ≤ 1,we obtain from (12)

‖B̂(N) − B̂(Ñ)‖ ≤12C

I∑

i=1

i∑

j=1

∑

xi≤xj+xk

Now we apply a useful equality NjNk − ÑjÑk = 12 [(Nj + Ñj)(Nk − Ñk) + (Nj − Ñj)(Nk + Ñk)]to get

‖B̂(N) − B̂(Ñ)‖ ≤ 12C

I∑

j=1

I∑

k=1

[

|(Nj + Ñj)||(Nk − Ñk)| + |(Nj − Ñj)||(Nk + Ñk)|]

. (33)

It can be easily shown that the total number of particles decreases in a coagulation process, i.e.

I∑

j=1

Nj ≤ N0T := Total number of particles which are taken initially.

The equation (33) can be rewritten as

‖B̂(N) − B̂(Ñ)‖ ≤ N0T C[ I∑

k=1

|(Nk − Ñk)| +I∑

j=1

|(Nj − Ñj)|]

≤ 2N0T C‖N− Ñ‖. (34)

Now we consider the death term

‖D̂(N) − D̂(Ñ)‖ =I∑

i=1

|D̂i(N) − D̂i(Ñ)|

≤I∑

i=1

I∑

j=1

β(xi, xj)|NiNj − ÑiÑj |

≤ CI∑

i=1

I∑

j=1

|NiNj − ÑiÑj|.

Again we use the same equality as before to get

‖D̂(N) − D̂(Ñ)‖ ≤ 2N0T C‖N − Ñ‖. (35)

So we can apply Theorem 2.3 to check the positivity of the solution obtained by the fixed pivottechnique.

Proposition 5.1. The numerical solution by the fixed pivot technique is non-negative.

Proof. The Lipschitz condition on F̂(t,M̂) can be easily shown by using (34), (35) and (7). Thedeath term D̂i in (12) will become zero when M̂i = 0 and (7) gives F̂i(t,M̂) ≥ 0. Thus, theTheorem 2.3 directly implies the non-negativity of the solution.

Now we shall prove the following convergence theorem.

Theorem 5.2. Let us assume that the Lipschitz conditions on B̂(N(t)) and D̂(N(t)) are satisfiedfor 0 ≤ t ≤ T and for all N, N̂ ∈ RI where N and N̂ are the projected exact and numericalsolutions defined in (3) and (7) respectively. More precisely, there exists a Lipschitz constantL := 2N0T C < ∞ such that (34) and (35) hold. Then a consistent discretization method is alsoconvergent and the convergence is of the same order as the consistency.

22

We need the following Gronwall Lemma to prove the above theorem. A slightly more generalresult is given in Linz [27]. For completeness we give the short proof.

Lemma 5.3. If v(t) satisfies

|v(t)| ≤ k∫ t

0|v(τ)|dτ +

∫ t

0|r(τ)|dτ, (36)

with k > 0,

max0≤t≤T

|r(t)| ≤ R > 0,

then

|v(t)| ≤ Rk

[exp (kt) − 1]. (37)

Proof. Let z(t) be the solution of

z(t) = k

∫ t

0z(τ)dτ + tR.

Since z(t) is a positive and increasing function of t

z(t) ≥ k∫ t

0z(τ)dτ +

∫ t

0|r(τ)|dτ,

and comparing this with (36) we have

z(t) ≥ |v(t)|.

But

z(t) =R

k[exp (kt) − 1].

and (37) follows.

Proof of Theorem 5.2

Using the equations (9) and (10) we have for ǫ(t) = N(t) − N̂(t)

d

dtǫ(t) = σ(t) − (B̂(N) − B̂(N̂)) + (D̂(N) − D̂(N̂)).

We then take the norm on both sides to get

d

dt‖ǫ(t)‖ ≤ ‖σ(t)‖ + ‖(B̂(N) − B̂(N̂))‖ + ‖(D̂(N) − D̂(N̂))‖.

Integrating with respect to t with ǫ(0) = 0 and using the Lipschitz conditions (34)-(35) weobtain the estimates

‖ǫ(t)‖ ≤∫ t

0‖σ(τ)‖dτ + 2L

∫ t

0‖ǫ(τ)‖dτ.

23

Table 1: Uniform grids

Grids Points Relative Error L1 EOC

200 - -400 0.0598 -800 0.0178 1.75191600 5.0E-3 1.82843200 1.3E-3 1.9528

From this it follows by Lemma 5.3 that

‖ǫ(t)‖ ≤ eh2L

[exp (2Lt) − 1], (38)

where

eh = max0≤t≤T

‖σ(t)‖.

If the scheme is consistent then limh→0

eh = 0. This completes the proof of the Theorem 5.2.

Remark 5.4. The proof of the Theorem 5.2 follows a related result by Linz [27].

6 Numerical Examples

We now justify our mathematical results on the convergence by taking a few numerical exampleswhere we evaluate the experimental order of convergence (EOC) numerically.Now we have to use a suitable ODE solver to solve the resulting set of ODEs. Integrating theresultant system (12) by using a standard ODE routine, for example the ODE45, ODE15S solversin MATLAB, may lead to negative values for the number density at large size classes. Thesenegative values may lead in consequence to instabilities in the overall computation. Therefore,one should take care of the positivity of the solution by the numerical integration routine.First of all we calculate the EOC for uniform meshes. For a uniform mesh, we consider thefollowing normally distributed initial condition

n(0, x) =1

σ√

2πexp

[

−(x − µ)2

2σ2

]

. (39)

In addition, we take the following aggregation kernel

β(x, y) = β0(x + y). (40)

Since analytical solutions are not available for the above initial condition and aggregation kernel,we use the following formula in order to calculate the experimental order of convergence

EOC = ln

(

‖N̂h − N̂h/2‖‖N̂h/2 − N̂h/4‖

)

/

ln(2). (41)

24

Table 2: Non-uniform smooth grids


60 6.4E-3 -120 1.6E-3 1.98240 4.0E-4 1.98480 1.0E-4 1.99

Table 3: Locally uniform grid


60 0.0303 -120 0.0156 0.96240 7.7E-3 1.02480 3.8E-3 1.03

Here N̂h represents the numerical solution on a uniform mesh of width h. For the numericalcomputation we have taken the minimum and maximum values of x as 0 and 50 respectively.The other parameters are σ2 = 0.01, µ = 1 and β0 = 1. The computation time t is set tobe 0.3567. The numerical results are shown in Table 1. As expected from the mathematicalanalysis, the numerical results exhibit convergence of second order.

Now we consider the second case of non-uniform smooth meshes. As mentioned before, sucha mesh can be obtained by some smooth transformation on a uniform mesh. Here we haveconsidered the exponential transformation as x = exp(ξ), where ξ is the variable for which wehave the uniform mesh. Such a mesh is also known as a geometric mesh. In this case we take anexponentially decreasing function, namely exp(−x) as initial condition. The linear aggregationkernel (40) is considered and the computation time is t = 0.3567. The computational domainin this case is taken as [1e − 6, 1000] which corresponds to the ξ domain [ln(1e − 6), ln(1000)].It should be pointed out that any small positive real number can be chosen as the minimumvalue of x. The analytical solution for this case can be found in Aldous [1]. Since the analyticalsolution is known, we can calculate the error EI on a mesh with I cells. Then the experimentalorder of convergence can be determined by the formula

EOC = ln(EI/EpI)/ ln(p), (42)

where EI and EpI are the L1 error norms. The subscripts I and pI correspond to the degreesof freedom. In this case we take p = 2 to evaluate the EOC. The numerical results have beensummarized in Table 2. The relative error has been calculated by dividing the error ‖N−N̂‖ by‖N‖. Once again the numerical results show that the fixed pivot technique gives second orderconvergence on non-uniform smooth meshes.

The third test case has been performed on a locally uniform mesh using the same problemand the same computational parameters as in the previous case. In this case we started the

25

Table 4: Oscillatory grids


60 0.0928 -180 0.0937 -0.0093540 0.0933 0.00431620 0.0910 0.0226

Table 5: Non-uniform random grids


120 0.0178 -240 0.0288 -0.6983480 0.0285 -0.3190960 0.0294 -0.2488

computation on 30 geometric mesh points, and then each cell was divided into two equal partsin the further refines levels of computation. In this way we obtained a locally uniform mesh.The EOC has been summarized in Table 3. As expected, the table clearly shows that the fixedpivot technique is only first order accurate.

Now let us consider the fourth case of an oscillatory mesh to evaluate the EOC. We considerthe same problem as for the third and second case. For the numerical computation we havetaken the minimum and maximum values of x as 0.001 and 18.001 respectively. First we dividedthe computational domain in 60 oscillatory mesh points and performed the computations forthe same. Here we did not make further refinement of meshes. It means for the next levelsagain we discretized the domain in the oscillatory mesh points of values I = 180, 540, 1620 andperformed the computations separately. To evaluate the EOC we consider p = 3 in this caseonly. The numerical results has been summarized in Table 4. Table 4 exhibits that the fixedpivot technique is not convergent on oscillatory meshes.Finally we consider the fifth case of a non-uniform random mesh. The computations have beenperformed on the same problem as for the third and second cases. We started again with the60 geometric mesh points, and then each cell was divided into two parts of random width in thefurther refined levels of computation. For each value of I = 120, 240, 480, 960, we performed fiveruns on different random grids and the relative L1 errors were measured. The mean of theseerrors over five runs is used to calculate the EOC. The numerical results have been shown inTable 5. Table 5 shows clearly that the fixed pivot technique is not convergent.

7 Conclusions

A detailed study on the convergence analysis of the fixed pivot technique is given for solvingaggregation PBE. It is remarked that the order of convergence depends on the type of themeshes chosen for the computation. The technique is second order convergent on uniform and

26

non-uniform smooth meshes. However, it is only first order convergent on locally uniform meshes.Finally, the scheme is examined closely on oscillatory and non-uniform random meshes and itis observed that the technique is not convergent. Furthermore, all observations are verifiednumerically.

Acknowledgments

This work was supported by the International Max-Planck Research School, ’Analysis, De-sign and Optimization in Chemical and Biochemical Process Engineering’, Otto-von-Guericke-Universität Magdeburg. The authors gratefully thank for funding of A.K. Giri through this PhDprogram.

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IntroductionThe Sectional MethodsThe Fixed Pivot TechniqueConsistencyOrder of the birth termBirth term on internal cellsBirth term on boundary cells

Order of the death term and summary of all termsMeshesUniform meshNon-uniform smooth meshLocally uniform meshOscillatory meshNon-uniform random mesh

ConvergenceNumerical ExamplesConclusions

Convergence analysis of sectional methods for solving ...population balance equations: The ﬁxed...

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