**5. Monotonization algorithm for the particle method**

#### **5.1 One-dimensional case**

Let us discuss the concept of the algorithm as applied to a one-dimensional flow for a single particle migrating from cell to cell (**Figures 1** and **2**). The figures show two cells containing particles represented by dots and imaginary boundaries of volumes represented by dashed segments. Note that calculations by this technique require only numerical values of the volumes, not their layout.

The flow is directed from left to right as indicated by velocity vector (**Figure 1**). Volume ΔV flowing out of the left cell (in what follows we call it the *volume flux*, darker color) is then equal to the product of the cell's lateral side length L and quantity S = u�τ:

$$
\Delta \mathbf{V} = \mathbf{L} \cdot \mathbf{u} \cdot \mathbf{\tau}. \tag{9}
$$

The non-monotonic behavior of the classical PIC method stems from the discrepancy between the real volume flux (and, accordingly, the mass flux) calculated by (9) and the volume of the particle crossing the cell side. In one case (**Figure 1a**), the volume moving from the left cell is smaller than the particle volume, and in the

**Figure 1.**

*Illustration of the reason for the non-monotonic behavior: a) volume flux is smaller than the particle volume; b) volume flux is larger than the particle volume.*

other (**Figure 1b**), it is larger than the particle volume. In the 2D case and in the case of several migrating particles, the situation remains fundamentally the same.

Let us introduce the following notation:

$$
\delta \mathbf{V} = \Delta \mathbf{V} \cdot \mathbf{V\_p},\tag{10}
$$

1.*The volume flux* ΔV *is smaller than the migrating particles' total volume* (*δ*V < 0). In this case, to be split are all the migrating particles. The mother particles

> X p Vp

*:* (12)

*<sup>p</sup> :* (13)

!

migrating into the acceptor cell have volumes

volume each:

*A Monotonic Method of Split Particles DOI: http://dx.doi.org/10.5772/intechopen.97044*

split first.

**5.2 Two-dimensional extension**

AF and BH, respectively.

**Figure 3.**

**49**

*Illustration of volume flux calculations in 2D case.*

provision is made for diagonal cell-to-cell fluxes.

V<sup>M</sup>

<sup>p</sup> ¼ Vp ΔV*=*

The daughter particles stay in the donor cell and have the following

*<sup>p</sup>* <sup>¼</sup> <sup>V</sup>*<sup>p</sup>* � <sup>V</sup>*<sup>M</sup>*

2.*Volume flux ΔV is larger than the migrating particles' total volume* (δV > 0). The particle staying next to the interface on the donor side is split to fill the remaining volume δV. Note that if Vp < δV, the mother particle's volume entirely goes to the daughter particles, and to be split is the next particle from the donor cell. If the donor cell is mixed, and the acceptor cell is pure and filled with the material present in the donor cell, the particles of this material will be

Of particular interest in this case is the particle transition to the neighbor cell located diagonally from the donor cell (**Figure 3**). This case is special, because EGAK solves the advection equation using decomposition in directions, whereas no

Consider the particular case depicted in **Figure 3**. Suppose only one node A moves to the new position B in the Eulerian step. The dashed lines in the figure show the locations of the cell sides, for which the grid node is a common vertex, at tn+1/2. C and D denote the points of intersection of straight lines AG and BE, and

V*<sup>D</sup>*

where ΔV is the volume flux calculated by (4), VP is the volume of particle p migrating from one cell into another.

Below we explain the monotonization algorithm for both of these cases.


If more than one cell migrates from cell to cell, formula (10) will take the form of

$$
\delta \mathbf{V} = \Delta \mathbf{V} - \sum\_{\mathbf{p}} \mathbf{V}\_{\mathbf{p}},
\tag{11}
$$

where ΔV is the volume flux calculated by (9); VP is the volume of the particle with index p migrating from cell to cell; summing is performed for all transferred particles.

Here, let us describe the differences from the algorithm described above.

**Figure 2.** *Illustration of the monotonization algorithm in the 1D case: a)* δ*V < 0; b)* δ*V > 0.* other (**Figure 1b**), it is larger than the particle volume. In the 2D case and in the case of several migrating particles, the situation remains fundamentally the same.

where ΔV is the volume flux calculated by (4), VP is the volume of particle p

1.*Volume flux* ΔV *is smaller than the migrating particle volume* (*δ*V < 0). This case is illustrated in **Figure 2a**. Left is the state at tnþ1*<sup>=</sup>*2; right, at tnþ1. In this case, the particle migrating from the donor to the acceptor cell splits into two parts, a mother and a daughter particle. The mother particle migrates into the acceptor cell and now has new coordinates corresponding to its velocity and a new volume equal to the volume flux leaving the donor cell ΔV. The daughter particle, whose volume is equal to the difference between the initial particle

Below we explain the monotonization algorithm for both of these cases.

volume and volume flux ΔV, is placed in the donor cell and acquires coordinates on the cell side. The link between the mother and the daughter

If more than one cell migrates from cell to cell, formula (10) will take the

*<sup>δ</sup>*<sup>V</sup> <sup>¼</sup> <sup>Δ</sup><sup>V</sup> �<sup>X</sup>

Here, let us describe the differences from the algorithm described above.

*Illustration of the monotonization algorithm in the 1D case: a)* δ*V < 0; b)* δ*V > 0.*

p

where ΔV is the volume flux calculated by (9); VP is the volume of the particle with index p migrating from cell to cell; summing is performed for all transferred

Vp, (11)

2.*Volume flux* Δ*V is larger than the migrating particle volume* (*δ*V > 0). This case is illustrated in **Figure 2b**. In this case, the missing volume of the migrating particle must be made up by forced transfer of some particles or particle fragments from the donor cell to the acceptor. To be split is the particle lying next to the side of these cells and not yet transferred to the acceptor cell. It produces a daughter particle of volume δV, which migrates into the acceptor

<sup>δ</sup><sup>V</sup> <sup>¼</sup> <sup>Δ</sup>V‐Vp, (10)

Let us introduce the following notation:

particle is indicated by a broken line.

cell with donor-acceptor side coordinates.

form of

particles.

**Figure 2.**

**48**

migrating from one cell into another.

*Recent Advances in Numerical Simulations*

1.*The volume flux* ΔV *is smaller than the migrating particles' total volume* (*δ*V < 0). In this case, to be split are all the migrating particles. The mother particles migrating into the acceptor cell have volumes

$$\mathbf{V\_{p}^{M}} = \mathbf{V\_{p}} \left(\boldsymbol{\Delta}\mathbf{V} / \sum\_{\mathbf{P}} \mathbf{V\_{p}}\right). \tag{12}$$

The daughter particles stay in the donor cell and have the following volume each:

$$\mathbf{V}\_p^D = \mathbf{V}\_p - \mathbf{V}\_p^M. \tag{13}$$

2.*Volume flux ΔV is larger than the migrating particles' total volume* (δV > 0). The particle staying next to the interface on the donor side is split to fill the remaining volume δV. Note that if Vp < δV, the mother particle's volume entirely goes to the daughter particles, and to be split is the next particle from the donor cell. If the donor cell is mixed, and the acceptor cell is pure and filled with the material present in the donor cell, the particles of this material will be split first.

### **5.2 Two-dimensional extension**

Of particular interest in this case is the particle transition to the neighbor cell located diagonally from the donor cell (**Figure 3**). This case is special, because EGAK solves the advection equation using decomposition in directions, whereas no provision is made for diagonal cell-to-cell fluxes.

Consider the particular case depicted in **Figure 3**. Suppose only one node A moves to the new position B in the Eulerian step. The dashed lines in the figure show the locations of the cell sides, for which the grid node is a common vertex, at tn+1/2. C and D denote the points of intersection of straight lines AG and BE, and AF and BH, respectively.

**Figure 3.** *Illustration of volume flux calculations in 2D case.*

In EGAK, the cell-to-cell volume fluxes are defined as follows. The volume flux corresponding to triangle ABG (for short, *volume in triangle ABG*) is transferred from cell (i, k) to cell (i + 1, k), that is:

$$\begin{aligned} V\_{i,k}^{n+1} &= V\_{i,k}^{n+1/2} - V\_{ABG}, \\ V\_{i+1,k}^{n+1} &= V\_{i+1,k}^{n+1/2} + V\_{ABG}. \end{aligned} \tag{14}$$

• One mother particle may have several daughter particles;

• One daughter particle may have only one mother particle;

will "remember" the index of their initial mother particle;

**6. Algorithm of interaction between particles and particle-free**

cells containing materials with and without particles.

splitting algorithm for the case of δV > 0 from Section 4.

**materials**

*A Monotonic Method of Split Particles DOI: http://dx.doi.org/10.5772/intechopen.97044*

Section 4, case δV < 0).

**7. Near-interface algorithms**

*Particle layout near the interface: a) t = 0, b) t > 0.*

**Figure 5.**

**51**

• The daughter particles may also split, but all the subsequent daughter particles

• The daughter particles are placed at the common donor/acceptor side to ensure their quickest possible combination with their respective mother particles.

The algorithm for volume flux calculations has a modification to deal with mixed

Consider the case of a cell filled with heterogeneous materials, one described

Otherwise, the missing part of the outflowing volume flux is filled with the particle-free material. If there are several particle-free materials, the volume is distributed among them based on the VOF algorithm [1]. If the particle-free materials are still not enough, the remaining volume flux is filled with particles using the

As part of the proposed method, we have developed an algorithm involving the particles located only near the interface. The region near the interface includes mixed cells and one layer of adjacent pure cells of each material on each side.

**Figure 5** shows possible particle layouts relative to the interface. The dark and light cells (**Figure 5a**) are pure, and the intermediate-color cell (**Figure 5b**) is

only by grid quantities, and the other, by particles (1 and 2, respectively, in **Figure 4**). Suppose we need to divide the flux moving from left to right (shown with a darker color) between the materials. The volume flux leaving the cell is first filled with the volume of migrating particles. If the migrating particles' total volume exceeds this volume flux, then the above particle splitting algorithm is engaged (see

Accordingly, the following relationships apply to all the cells under consideration including all the fluxes:

$$\begin{aligned} V\_{i,k}^{n+1} &= V\_{i,k}^{n+1/2} - V\_{ABG} - V\_{ABF} = V\_{BFKG}, \\ V\_{i+1,k}^{n+1} &= V\_{i+1,k}^{n+1/2} + V\_{ABG} - V\_{ABE} = V\_{BGNE}, \\ V\_{i,k+1}^{n+1} &= V\_{i,k+1}^{n+1/2} - V\_{ABH} + V\_{ABF} = V\_{BFLH}, \\ V\_{i+1,k+1}^{n+1} &= V\_{i+1,k+1}^{n+1/2} + V\_{ABE} + V\_{ABH} = V\_{BHME}. \end{aligned} \tag{15}$$

Note that in accordance with (15) the volume of triangle ABC is included in the volume of cell (i + 1,k) twice – as part of triangles ABG and ABE – but in one case it is positive, and in the other, negative. Thus, in fact it is not included in the updated volume of this cell; but it will be included in the volume of cell (i + 1,k + 1). The same applies to the volume of triangle ABD, which will be included in the volume of cell (i + 1,k + 1) and not included in the volume of cell (i,k + 1).

In accordance with the above, when considering particle contributions, flux calculations between cell (i,k) and its non-diagonal neighbors assume that the particle lying in triangle ABC migrates into cell (i + 1,k), and the particle lying in triangle ABD, into cell (i,k + 1). Then, in calculations of the flux between cells (i + 1, k), (i + 1,k + 1) and (i,k + 1), (i + 1,k + 1), these particles migrate into cell (i + 1, k + 1). Therefore, when considering this process in terms of flux monotonicity, corresponding daughter particles are introduced as shown in **Figure 4**. Note that the mother particle's position during the flux calculations is nevertheless defined in the true acceptor cell (i + 1,k + 1).

The particle splitting is based on the following principles:


**Figure 4.** *Volume flux from a mixed cell.*

In EGAK, the cell-to-cell volume fluxes are defined as follows. The volume flux corresponding to triangle ABG (for short, *volume in triangle ABG*) is transferred

*<sup>i</sup>*,*<sup>k</sup>* � *VABG*,

*<sup>i</sup>*þ1,*<sup>k</sup>* <sup>þ</sup> *VABG:* (14)

(15)

*Vn*þ<sup>1</sup>

*Vn*þ<sup>1</sup>

*<sup>i</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>V</sup>n*þ1*=*<sup>2</sup>

*<sup>i</sup>*þ1,*<sup>k</sup>* <sup>¼</sup> *<sup>V</sup>n*þ1*=*<sup>2</sup>

Accordingly, the following relationships apply to all the cells under consider-

*<sup>i</sup>*,*<sup>k</sup>* � *VABG* � *VABF* ¼ *VBFKG*,

*<sup>i</sup>*þ1,*<sup>k</sup>* <sup>þ</sup> *VABG* � *VABE* <sup>¼</sup> *VBGNE*,

*<sup>i</sup>*,*k*þ<sup>1</sup> � *VABH* <sup>þ</sup> *VABF* <sup>¼</sup> *VBFLH*,

Note that in accordance with (15) the volume of triangle ABC is included in the volume of cell (i + 1,k) twice – as part of triangles ABG and ABE – but in one case it is positive, and in the other, negative. Thus, in fact it is not included in the updated volume of this cell; but it will be included in the volume of cell (i + 1,k + 1). The same applies to the volume of triangle ABD, which will be included in the volume of

In accordance with the above, when considering particle contributions, flux calculations between cell (i,k) and its non-diagonal neighbors assume that the particle lying in triangle ABC migrates into cell (i + 1,k), and the particle lying in triangle ABD, into cell (i,k + 1). Then, in calculations of the flux between cells (i + 1, k), (i + 1,k + 1) and (i,k + 1), (i + 1,k + 1), these particles migrate into cell (i + 1, k + 1). Therefore, when considering this process in terms of flux monotonicity, corresponding daughter particles are introduced as shown in **Figure 4**. Note that the mother particle's position during the flux calculations is nevertheless defined in

• Both particles produced from the particle being split share its thermodynamic

• The index assigned to the daughter particle is the same as the index of its mother particle, which also indicates that the particle is a daughter;

• The mother particle "knows" nothing about its daughter particles;

*<sup>i</sup>*þ1,*k*þ<sup>1</sup> <sup>þ</sup> *VABE* <sup>þ</sup> *VABH* <sup>¼</sup> *VBHME:*

from cell (i, k) to cell (i + 1, k), that is:

*Recent Advances in Numerical Simulations*

*Vn*þ<sup>1</sup>

*Vn*þ<sup>1</sup>

*Vn*þ<sup>1</sup>

*V<sup>n</sup>*þ<sup>1</sup>

the true acceptor cell (i + 1,k + 1).

**Figure 4.**

**50**

*Volume flux from a mixed cell.*

*<sup>i</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>V</sup>n*þ1*=*<sup>2</sup>

*<sup>i</sup>*þ1,*<sup>k</sup>* <sup>¼</sup> *<sup>V</sup>n*þ1*=*<sup>2</sup>

*<sup>i</sup>*,*k*þ<sup>1</sup> <sup>¼</sup> *<sup>V</sup><sup>n</sup>*þ1*=*<sup>2</sup>

*<sup>i</sup>*þ1,*k*þ<sup>1</sup> <sup>¼</sup> *<sup>V</sup><sup>n</sup>*þ1*=*<sup>2</sup>

cell (i + 1,k + 1) and not included in the volume of cell (i,k + 1).

The particle splitting is based on the following principles:

state (to comply with the conservation laws);

ation including all the fluxes:

