Convergence and error analysis for pure collisional breakage equation

Particulate processes are prominent in the dynamics of particle development and describe how particles might unite to generate larger ones or break into smaller ones. Suppose that each particle is entirely defined by a single size variable, such as its volume or mass. Particle breakage is categorized into linear breakage and collision breakage. The linear breakage equation’s success is well-known in investigating phenomena of importance in various scientific areas ranging from engineering, see [1, 2] and further citations. It is believed that its expansion is required to broaden the variety of procedures that may be evaluated and to increase analysis quality. One conceivable expansion is to include nonlinearity in the breaking process which can occur when the breakage behaviour of a particle is determined not only by its characteristics and dynamic circumstances (as in linear breakage), but also by the state and properties of the entire system, i.e., by binary interactions, collisional breakage could enable some mass transfer between colliding particles. As a result, daughter particles with more extensive volumes than the parent particles are generated. Non-linear models emerge in a wide range of contexts, including milling and crushing processes [3, 4], bulk distribution of asteroids [5, 6], fluidized beds [7, 8], etc.
Cheng and Redner [9] used the following integro-differential equation to derive the collisional breakage equation (CBE). It illustrates the time progession of particle size distribution $c(t,x)\geq 0$ of particles of mass $x\in]0,\infty[$ at time $t\geq 0$ and is defined by

with the given initial data

The characteristics $t$ and $x$ are regarded as dimensionless quantities without losing any generality. In Eq.(1), the collision kernel $K(x,y)$ depicts the rate of collision for breakage event between two particles of volumes $x$ and $y$. In practice, it is assumed that the collision rate between the particles of volumes $x$ and $y$ is identical to the collision between $y$ and $x$. The term $b(x,y,z)$ is called the breakage distribution function, which defines the rate for production of particles of volume $x$ by breakage of particle of volume $y$ due to interaction between particles of volumes of $y$ and $z$. Breakage distribution function $b$ holds

as well as satisfies

for all $(y,z)\in{]0,\infty[}^{2}$. The first term in Eq.(1) explains gaining particles of volume $x$ due to collision between particles of volumes $y$ and $z$, known as the birth term. The second term is labeled as the death term and describes the disappearance of particles of volume $x$ due to collision with particles of volume $y$.

It is also necessary to specify some integral features of the number density function $c(t,x)$, known as moments. The following equation defines the $j^{th}$ moment of the solution as

The zeroth and first moments are proportional to the total number of particles in the system and its total volume, respectively. Here, $M_{1}^{in}$ denotes the initial volume of the particles in the closed particulate system.

Before venturing into the specifics of the current work, let us review the existing literature on the linear breakage equation that has been widely investigated over the years concerning its analytical solutions [10], similarity solutions [11, 12], numerical results [13, 14].

There is a substantial body of work on the well-posedness of the coagulation and linear breakage equations (CLBE). The authors examined the existence and uniqueness of solutions to CLBE with nonsingular coagulation kernels with different growth parameters on the breakage function in [15, 16, 17]. Moreover, many publications are accessible for solving coagulation-fragmentation equations numerically, including the method of moments [18], finite element scheme [19], Monte Carlo methodology [20], and finite volume method (FVM)[21, 22, 23] to name a few. It has been reported by several authors that the FVM is an appropriate option  among the other numerical techniques for solving coagulation-fragmentation equations due to its mass conservation property.

Collisional breakage model is discussed in just a few mathematical articles in the literature, see [24, 25, 26]. The authors explained the global classical solutions of coagulation and collisional equation with collision kernel, growing indefinitely for large volumes in [26]. In addition, fewer publications in the physics literature [27, 28, 29] are committed to the collision breakage equation, with the majority dealing with scaling behavior and shattering transitions. Laurençot and Wrzosek [24] investigated the existence of a solution for discrete collisional breakage with coagulation equation in which they have used the following constraint over the kernels

They have also explored gelation and the long-term behavior of solutions. Further, in [30], the analysis is worked out with the coagulation dominating process for mass conserving solutions when the collision kernel grows at most linearly at infinity. To the best of the authors’ knowledge, none of the prior studies account for the weak convergence of the numerical scheme for solving collisional breakage equation. Therefore, this article is an attempt to study the weak convergence analysis of the model for nonsingular unbounded kernels in a numerical sense and then error estimation for kernels in $W_{loc}^{1,\infty}$ space over a uniform mesh. Thanks to the idea taken from Bourgade and Filbet [22], in which they have treated coagulation and binary fragmentation equation. The proof is based on the weak $L^{1}$ compactness method.

To proceed further, firstly, we will concentrate on the functional setting as having in mind that expected mass conservation in (3) is necessary. Besides the first moment, the total number of particles in the system must be finite. Therefore, we construct the solution space that exhibits the convergence of the discretized numerical solution to the weak solution of the collisional breakage equation (1), which is recognized as a weighted $L^{1}$ space such as

where $\|c\|=\int_{0}^{\infty}(1+x)c(x)\,dx,$ for the non-negative initial condition $c^{in}\in X^{+}$ and $\mathbb{R}^{+}=]0,\infty[.$ Here the notation $L^{1}({\mathbb{R}}^{+},xdx)$ stands for the space of the Lebesgue measurable real-valued functions on $\mathbb{R}^{+}$ which are integrable with respect to the measure $x\,dx.$

Now, the specifications of the collisional kernel $K$ and breakage distribution function $b$ are expressed in the following expression: both functions are symmetric and measurable over the domain. There exist $\zeta,\eta$ with $0<\zeta\leq\eta\leq 1,\,\zeta+\eta\leq 1$ and $\alpha\in\mathbb{R}$, $\lambda>0$ such that

H2:

This article’s contents are organized as follows. The discretization methodology based on the FVM and non-conservative form of fully discretized CBE are introduced in Section 2, followed by the detailed convergence analysis in Section 3. In Section 4, we examine the first order error estimates of FVM on uniform meshes. Additionally, we have justified the theoretical error estimation via numerical results in section 5. Consequently, in the final Section, some conclusions are presented.

In this section, we commence exploring the FVM for the solution of Eq.(1). It is based on the spatial domain being divided into tiny grid cells. Particle volumes ranging from $0$ to $\infty$ are taken into account in Eq.(1). Nevertheless, we define the particle volumes to be in a finite domain for practical purposes. Consider the reduced computational domain for volumes (0,$R$], with $0<R<\infty$. Thus the collisional breakage equation is truncated as

with the given initial distribution

Consider a partitioning of the operating domain $(0,R]$ into small cells as $\Lambda_{i}^{h}:=]x_{i-1/2},x_{i+1/2}],\,\,i=1,2,...,\mathrm{I}$, where,   $x_{1/2}=0,\ \ x_{\mathrm{I}+1/2}=R,\hskip 5.69046pt\Delta x_{i}=x_{i+1/2}-x_{i-1/2}$ and consider $h=max\,\Delta x_{i}\ \forall\ i.$ The grid points are the midpoints of each subinterval and are designated as

Now, the expression of the mean value of the number density function $c_{i}(t)$ in the cell $\Lambda_{i}^{h}$ is determined by

where $\Delta x_{i}=x_{i+1/2}-x_{i-1/2}$ for $i=1,2,...,\mathrm{I}.$ The domain is confined in the range [0, T] for the time parameter, and it is discretized into $N$ time intervals with time step $\Delta t$. The interval is defined as

We now begin developing the scheme on non-uniform meshes. It has the significant advantage of allowing the inclusion of a more extensive domain with fewer mesh points than a uniform mesh. The discretization differs slightly from that of Filbet and Laurencot [22], where they first converted the model (1) to a conservative equation using Leibniz integral rule, then discretized using FVM. Although, in this work, we have developed a non-conservative scheme using FVM from the continuous equation (1).

To derive the discretized version of the CBE (6), we proceed as follows: integrate the Eq.(6) with respect to $x$ over $i^{th}$ cell yields the following discrete form

where

along with initial distribution,

Implementing the midpoint rule to all of the above representation yields the semi-discrete equation after some simplifications as

where the term $p_{j}^{i}$ is expressed by

Now, to obtain a fully discrete system, applying explicit Euler discretization to time variable $t$ leads to

For the convergence analysis, consider a function $c^{h}$ on $[0,T]\times]0,R]$ which is representated by

where $\chi_{D}(x)$ denotes the characteristic function on a set $D$ as $\chi_{D}(x)=1$ if $x\in D$ or $0$ everywhere else. Also noting that

converges strongly to $c^{in}$ in $L^{1}((0,R))$ as $h\rightarrow 0$. A finite volume approximation approaches the kernels on each space cell, i.e., for all $(u,v)\in]0,R]\times]0,R]$ and $(u,v,w)\in]0,R]\times]0,R]\times]0,R],$

where

Such discretization ensures that $\|K^{h}-K\|_{L^{1}((0,R)\times(0,R))}\rightarrow 0$, $\|b^{h}-b\|_{L^{1}((0,R)\times(0,R)\times(0,R))}\rightarrow 0$ as $h\rightarrow 0$, see [22].

The objective of this section is to study the convergence of solution $c^{h}$ to a function $c$ as $h$ and $\Delta t$ $\rightarrow 0$.

Following the preceding theorem, it is evident that the main motivation here is to demonstrate the weak convergence of the family of functions $(c^{h})$ to a function $c$ in $L^{1}(0,R)$ as $h$ and $\Delta t$ approach zero. The idea is based on the Dunford-Pettis theorem, which establishes a necessary and sufficient condition for $L^{1}$ compactness in the presence of weak convergence.

As a result, demonstrating the equiboundedness and equiintegrability of the family $c^{h}$ in $L^{1}$ as in (20) and (21), respectively, is sufficient to establish Theorem 3.1. The following proposition addresses the non-negativity and equiboundedness of the functions $c^{h}$. For the proof, we employed Bourgade and Filbet’s approach [22].

To demonstrate the family of solutions’s uniform integrability, let us designate a specific category of convex functions as $C_{VP,\infty}$. Consider $\Phi\in C^{\infty}([0,\infty))$, a non-negative and convex function that belongs to the ${C}_{VP,\infty}$ class and has the following properties: