**1. Introduction**

Correct calculations of multi-material flows is the greatest challenge for ALE and Eulerian CFD codes, especially those using mixed cells at interfaces. There are two basic approaches to solving the advection equation for the multi-material case. In the first (grid-based) approach, interfaces are identified, and their position on the grid is tracked at each time step. The interface can be identified both explicitly, or it can be recovered based on the field of volume fractions. The latter algorithm serves as a basis for widely used methods, like the VOF method [1] (concentration method [2]). The second approach involves material particle methods first proposed by Harlow (the PIC method [3]). In this case, material fluxes from cells, including

mixed ones, are controlled by particles, to which certain material masses are assigned.

Both approaches have their advantages and drawbacks. The advantages of the particle method consist in the Lagrangian representation of particles and the possibility of assigning material information to them. This minimizes the errors of solving the advection equation by the grid-based Eulerian methods. A number of modifications of the PIC method have been developed to improve its accuracy and extend the range of its applications [4–9]. An overview of these methods is provided in [10].

(it generally coincides with the grid of the previous timestep), and the quantities are remapped onto it. As inputs, this step takes the outputs of the Lagrangian step. In turn, this step is divided into two sub-steps, i.e., approximation of the advection

The difference equations below are presented in as much detail as needed to understand the algorithm of particle introduction; please refer to [14, 15] for a more detailed description of EGAK's basic difference scheme. In what follows, if no confusion is possible, no subscripts or subscript *n* are used to denote the outputs of the previous timestep, and subscripts n+1/2 and n + 1 denote the outputs of the

The Lagrangian gas dynamic equations are approximated using EGAK's standard scheme. As outputs, the Lagrange step delivers updated node-centered velocities, as well as densities, energies and volume fractions of each constituent material. This

In addition to the material-specific quantities, some particle-specific quantities

*Step 1.* At the Lagrangian step, particles are assumed to move together with the cell and inside the cell, without crossing its boundaries. The relative change in the particle position in the cell is associated with the difference in divergences (compression ratios) of different materials as a result of employing one closing model or another for the mixed-cell gas dynamic equations. In this study, we use only one assumption that the materials have equal divergences. This means that the sub-cell motion of particles does not change their position relative to the grid nodes.

*Step 2*. It is easy to show that the calculations of particle velocities by bilinear

x1 <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

y1 <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

Here, М is the cell mass and uxi, uyi (i = 0, 1, 2, 3) are the velocity vector

y3 � <sup>M</sup>*:*

interpolation violate the law of conservation of momentum in the particlecontaining cell. To ensure its conservation, the calculated particle velocities are

x0 <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

y0 <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

<sup>p</sup> are updated by bilinear interpolation between

.

x2 <sup>þ</sup> unþ1*=*<sup>2</sup> x3 � M, (1)

y2 <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

equation is done with decomposition in directions.

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

Lagrangian and the Eulerian step, respectively.

**3.1 Approximation of the cell-centered quantities**

also applies to cells containing particles.

**3.3 Updating of particle coordinates**

Particle coordinates, x~<sup>n</sup>þ1*=*<sup>2</sup>

corrected as follows:

**45**

**3.2 Definition of particle-specific quantities**

are also defined for particles in the cells containing particles.

Updated particle coordinates are found in two steps:

<sup>p</sup> , y~<sup>n</sup>þ1*=*<sup>2</sup>

coordinates of cell nodes xn+1/2, yn+1/2, just as at time tn

1.Components of cell momentum are calculated:

<sup>4</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

<sup>4</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

Pcx <sup>¼</sup> <sup>1</sup>

Pcy <sup>¼</sup> <sup>1</sup>

components at four cell nodes.

**3. Lagrangian step**

The central drawback of the particle method is the highly non-monotonic character of the solution caused by the discrete transfer of mass and mass-related quantities from cell to cell. The corresponding error can be reduced in the most straightforward manner by increasing the number of particles in cells, but such an increase limits the method's performance, especially in the 3D case. To minimize this drawback, a number of method modifications are employed. In [11, 12], for this purpose, the authors use particles having different masses. This approach, however, does not eliminate the need of involving a large number of particles. In a different approach, particles are used only in a limited part of the integration domain, for example, near interfaces [13]. As a result, only a small number of cells contain large quantities of particles. The rest part of the domain in this case is treated by the gridbased methods. Such a selective use of particles, however, does not eliminate the error of solving the advection equation by the grid-based methods and the need of remembering the history of a particular process in a large volume of the material.

This paper proposes a particle method that minimizes these drawbacks. Monotonization of the particle method is performed by particle splitting, so that the material volume flowing out of the cell corresponds to the volume calculated by schemes based on the grid approach. In order to prevent endless splitting, such split particles are further recombined under certain conditions. This approach allows us to do with a small number of particles in the cell, while delivering a monotonic solution.

### **2. Problem statement**

The split-particle (SP) method has been implemented in a code called EGAK in the 2D approximation. In the source code, the major quantities for numerical solution of the multi-material gas dynamic equations include node-centered velocity vector components ux and uy and cell-centered thermodynamic quantities: density ρξ, specific (per unit mass) internal energy eξ, and volume fractions βξ = Vξ/V of the constituent materials.

Particles can also be specified for some materials (in a particular case, these can be all materials). Each particle (with index р) has its coordinates in space xр(t), yр(t) and velocity vector components uxр(t), uyр(t) (these are used in interim calculations in the Lagrangian step). In addition, all particles represent thermodynamic states of the corresponding material (density, specific internal energy, volume): ρξр, eξр, Vξр. Note that densities and volumes of particles can also give us their masses. Also note that in the method proposed particle velocities are obtained by interpolation between nodal velocities rather than "remembered" like in the classical PIC method.

Approximation of the corresponding equations is performed in two steps using a decomposition procedure. The **first** (Lagrangian) step involves calculations of the gas dynamic equations without convective members, i.e. gas dynamic equations in Lagrangian variables. In the **second** (Eulerian) step, a new grid is constructed

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

(it generally coincides with the grid of the previous timestep), and the quantities are remapped onto it. As inputs, this step takes the outputs of the Lagrangian step. In turn, this step is divided into two sub-steps, i.e., approximation of the advection equation is done with decomposition in directions.

The difference equations below are presented in as much detail as needed to understand the algorithm of particle introduction; please refer to [14, 15] for a more detailed description of EGAK's basic difference scheme. In what follows, if no confusion is possible, no subscripts or subscript *n* are used to denote the outputs of the previous timestep, and subscripts n+1/2 and n + 1 denote the outputs of the Lagrangian and the Eulerian step, respectively.

### **3. Lagrangian step**

mixed ones, are controlled by particles, to which certain material masses are

extend the range of its applications [4–9]. An overview of these methods is

The central drawback of the particle method is the highly non-monotonic character of the solution caused by the discrete transfer of mass and mass-related quantities from cell to cell. The corresponding error can be reduced in the most straightforward manner by increasing the number of particles in cells, but such an increase limits the method's performance, especially in the 3D case. To minimize this drawback, a number of method modifications are employed. In [11, 12], for this purpose, the authors use particles having different masses. This approach, however, does not eliminate the need of involving a large number of particles. In a different approach, particles are used only in a limited part of the integration domain, for example, near interfaces [13]. As a result, only a small number of cells contain large quantities of particles. The rest part of the domain in this case is treated by the gridbased methods. Such a selective use of particles, however, does not eliminate the error of solving the advection equation by the grid-based methods and the need of remembering the history of a particular process in a large volume of the material. This paper proposes a particle method that minimizes these drawbacks. Monotonization of the particle method is performed by particle splitting, so that the material volume flowing out of the cell corresponds to the volume calculated by schemes based on the grid approach. In order to prevent endless splitting, such split particles are further recombined under certain conditions. This approach allows us to do with a small number of particles in the cell, while delivering a monotonic

The split-particle (SP) method has been implemented in a code called EGAK in the 2D approximation. In the source code, the major quantities for numerical solution of the multi-material gas dynamic equations include node-centered velocity vector components ux and uy and cell-centered thermodynamic quantities: density ρξ, specific (per unit mass) internal energy eξ, and volume fractions βξ = Vξ/V of the

Particles can also be specified for some materials (in a particular case, these can be all materials). Each particle (with index р) has its coordinates in space xр(t), yр(t) and velocity vector components uxр(t), uyр(t) (these are used in interim calculations in the Lagrangian step). In addition, all particles represent thermodynamic states of the corresponding material (density, specific internal energy, volume): ρξр, eξр, Vξр. Note that densities and volumes of particles can also give us their masses. Also note that in the method proposed particle velocities are obtained by interpolation between nodal velocities rather than "remembered" like in the classi-

Approximation of the corresponding equations is performed in two steps using a decomposition procedure. The **first** (Lagrangian) step involves calculations of the gas dynamic equations without convective members, i.e. gas dynamic equations in Lagrangian variables. In the **second** (Eulerian) step, a new grid is constructed

Both approaches have their advantages and drawbacks. The advantages of the particle method consist in the Lagrangian representation of particles and the possibility of assigning material information to them. This minimizes the errors of solving the advection equation by the grid-based Eulerian methods. A number of modifications of the PIC method have been developed to improve its accuracy and

assigned.

solution.

**2. Problem statement**

constituent materials.

cal PIC method.

**44**

provided in [10].

*Recent Advances in Numerical Simulations*

#### **3.1 Approximation of the cell-centered quantities**

The Lagrangian gas dynamic equations are approximated using EGAK's standard scheme. As outputs, the Lagrange step delivers updated node-centered velocities, as well as densities, energies and volume fractions of each constituent material. This also applies to cells containing particles.

#### **3.2 Definition of particle-specific quantities**

In addition to the material-specific quantities, some particle-specific quantities are also defined for particles in the cells containing particles.

#### **3.3 Updating of particle coordinates**

Updated particle coordinates are found in two steps:

*Step 1.* At the Lagrangian step, particles are assumed to move together with the cell and inside the cell, without crossing its boundaries. The relative change in the particle position in the cell is associated with the difference in divergences (compression ratios) of different materials as a result of employing one closing model or another for the mixed-cell gas dynamic equations. In this study, we use only one assumption that the materials have equal divergences. This means that the sub-cell motion of particles does not change their position relative to the grid nodes.

Particle coordinates, x~<sup>n</sup>þ1*=*<sup>2</sup> <sup>p</sup> , y~<sup>n</sup>þ1*=*<sup>2</sup> <sup>p</sup> are updated by bilinear interpolation between coordinates of cell nodes xn+1/2, yn+1/2, just as at time tn .

*Step 2*. It is easy to show that the calculations of particle velocities by bilinear interpolation violate the law of conservation of momentum in the particlecontaining cell. To ensure its conservation, the calculated particle velocities are corrected as follows:

1.Components of cell momentum are calculated:

$$\mathbf{P}\_{\rm cx} = \frac{1}{4} \left( \mathbf{u}\_{\rm x0}^{\rm n+1/2} + \mathbf{u}\_{\rm x1}^{\rm n+1/2} + \mathbf{u}\_{\rm x2}^{\rm n+1/2} + \mathbf{u}\_{\rm x3}^{\rm n+1/2} \right) \cdot \mathbf{M},\tag{1}$$

$$\mathbf{P}\_{\rm cy} = \frac{1}{4} \left( \mathbf{u}\_{\rm y0}^{\rm n+1/2} + \mathbf{u}\_{\rm y1}^{\rm n+1/2} + \mathbf{u}\_{\rm y2}^{\rm n+1/2} + \mathbf{u}\_{\rm y3}^{\rm n+1/2} \right) \cdot \mathbf{M}.$$

Here, М is the cell mass and uxi, uyi (i = 0, 1, 2, 3) are the velocity vector components at four cell nodes.

2.Components of the total momentum of particles belonging to the cell are calculated:

$$\mathbf{P\_{px}} = \sum\_{\mathbf{P}} \tilde{\mathbf{u}}\_{\mathbf{xp}}^{\mathbf{n} + 1/2} \cdot \mathbf{m\_{p}}, \mathbf{P\_{py}} = \sum\_{\mathbf{P}} \tilde{\mathbf{u}}\_{\mathbf{pp}}^{\mathbf{n} + 1/2} \cdot \mathbf{m\_{p}} \tag{2}$$

In calculations, it is not always efficient to represent all constituent materials by particles because this requires extra calculations and computer memory. Therefore, it is reasonable to use particles for the constituent materials, for which the errors due to the solution of the advection equation are most essential, for example, for thin layers or materials, which require remembering the history of their Lagrangian

1. Interaction of materials described by particles and materials calculated by the

2. Support of existence, creation and removal of particles only in the vicinity of

These algorithms are listed in the order of their execution in the Eulerian step

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

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

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

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

ΔV ¼ L � u � τ*:* (9)

As part of the proposed SP method, the following algorithms have been

4.Remapping of particles density and energy to the cell as a whole.

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

require only numerical values of the volumes, not their layout.

particle.

developed:

code's standard scheme;

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

after the monotonization algorithm.

the interface;

3.Particles merging;

**5.1 One-dimensional case**

quantity S = u�τ:

**Figure 1.**

**47**

*volume flux is larger than the particle volume.*

Here, the particle velocities are calculated using the particles' coordinates determined by bilinear interpolation and previous particle coordinates:

$$
\tilde{\mathbf{u}}\_{\mathbf{x}\mathbf{p}}^{\mathbf{n}+\frac{1}{2}} = \frac{\tilde{\mathbf{x}}\_{\mathbf{p}}^{\mathbf{n}+1/2} - \mathbf{x}\_{\mathbf{p}}^{\mathbf{n}}}{\pi}, \tilde{\mathbf{u}}\_{\mathbf{y}\mathbf{p}}^{\mathbf{n}+\frac{1}{2}} = \frac{\tilde{\mathbf{y}}\_{\mathbf{p}}^{\mathbf{n}+1/2} - \mathbf{y}\_{\mathbf{p}}^{\mathbf{n}}}{\pi}, \tag{3}
$$

were τ = tn+1- tn .

3.Coefficients *λ*<sup>x</sup> ¼ Pcx*=*Ppx, *λ*<sup>y</sup> ¼ Pcy*=*Ppy are calculated.

The particles' velocities and coordinates are updated using the resulting weights:

$$\mathbf{u}\_{\mathbf{x}\mathbf{p}}^{\mathbf{n}+1/2} = \boldsymbol{\lambda}\_{\mathbf{x}} \cdot \ddot{\mathbf{u}}\_{\mathbf{x}\mathbf{p}}^{\mathbf{n}+1/2}, \mathbf{u}\_{\mathbf{y}\mathbf{p}}^{\mathbf{n}+1/2} = \boldsymbol{\lambda}\_{\mathbf{y}} \cdot \ddot{\mathbf{u}}\_{\mathbf{y}\mathbf{p}}^{\mathbf{n}+1/2};\tag{4}$$
 
$$\mathbf{x}\_{\mathbf{p}}^{\mathbf{n}+1/2} = \mathbf{x}\_{\mathbf{p}}^{\mathbf{n}} + \mathbf{u}\_{\mathbf{x}\mathbf{p}}^{\mathbf{n}+1/2} \cdot \boldsymbol{\tau}, \mathbf{y}\_{\mathbf{p}}^{\mathbf{n}+1/2} = \mathbf{y}\_{\mathbf{p}}^{\mathbf{n}} + \mathbf{u}\_{\mathbf{y}\mathbf{p}}^{\mathbf{n}+1/2} \cdot \boldsymbol{\tau}.$$

#### **3.4 Determination of particle velocity, density and energy**

Changes in the relative density and energy of particles of a given constituent material are assumed to be equal to the corresponding relative changes in these quantities calculated for the respective material on average. This gives the following formulas:

$$
\rho\_{\xi\mathbf{p}}^{\mathbf{n}+1/2} = \rho\_{\xi\mathbf{p}}^{\mathbf{n}} + \left(\rho\_{\xi}^{\mathbf{n}+1/2} - \rho\_{\xi}^{\mathbf{n}}\right)\rho\_{\xi\mathbf{p}}^{\mathbf{n}}/\rho\_{\xi}^{\mathbf{n}},\tag{5}
$$

$$\mathbf{e}\_{\xi\mathbf{p}}^{\mathbf{n}+1/2} = \mathbf{e}\_{\xi\mathbf{p}}^{\mathbf{n}} + \left(\mathbf{e}\_{\xi}^{\mathbf{n}+1/2} - \mathbf{e}\_{\xi}^{\mathbf{n}}\right),\tag{6}$$

$$\mathbf{V}\_{\xi\mathbf{p}}^{\mathbf{n}+1/2} = \mathbf{V}\_{\xi\mathbf{p}}^{\mathbf{n}} \left( \mathbf{V}\_{\xi}^{\mathbf{n}+1/2} / \mathbf{V}\_{\xi}^{\mathbf{n}} \right). \tag{7}$$

It is easy to show that, when using (5)–(7), the particles' total masses will remain unchanged, and the particles' total internal energies will be equal to the energy calculated for the given material as a whole, i.e. the following relationships hold:

$$\rho\_{\xi}^{\mathbf{n}+1/2}\mathbf{V}\_{\xi} = \sum\_{\mathbf{p}} \rho\_{\xi\mathbf{p}}^{\mathbf{n}+1/2} \mathbf{V}\_{\xi\mathbf{p}}, \mathbf{e}\_{\xi}^{\mathbf{n}+1/2}\mathbf{m}\_{\xi} = \sum\_{\mathbf{p}} \mathbf{e}\_{\xi\mathbf{p}}^{\mathbf{n}+1/2} \mathbf{m}\_{\xi\mathbf{p}}.\tag{8}$$

## **4. Eulerian step**

Major difficulties in implementing the particle method are associated with the Eulerian step, and namely, with calculations of mass and internal energy fluxes from cell to cell. In the PIC method, when a particle migrates to a neighbor cell, its mass and energy "migrate" with it. Because of the discrete (and, accordingly, nonmonotonic) character of the transfer of mass and all the quantities defined per unit mass, this method delivers highly non-monotonic quantity profiles. Section 4 provides a detailed description of the monotonization algorithm for the PIC method.

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

In calculations, it is not always efficient to represent all constituent materials by particles because this requires extra calculations and computer memory. Therefore, it is reasonable to use particles for the constituent materials, for which the errors due to the solution of the advection equation are most essential, for example, for thin layers or materials, which require remembering the history of their Lagrangian particle.

As part of the proposed SP method, the following algorithms have been developed:


2.Components of the total momentum of particles belonging to the cell are

xp � mp, Ppy <sup>¼</sup> <sup>X</sup>

Here, the particle velocities are calculated using the particles' coordinates determined by bilinear interpolation and previous particle coordinates:

> <sup>n</sup>þ<sup>1</sup> <sup>2</sup> yp <sup>¼</sup> <sup>y</sup>~nþ1*=*<sup>2</sup>

The particles' velocities and coordinates are updated using the resulting weights:

xp , u<sup>n</sup>þ1*=*<sup>2</sup>

xp � <sup>τ</sup>, ynþ1*=*<sup>2</sup>

Changes in the relative density and energy of particles of a given constituent material are assumed to be equal to the corresponding relative changes in these quantities calculated for the respective material on average. This gives the following

> nþ1*=*2 <sup>ξ</sup> � <sup>ρ</sup><sup>n</sup> ξ

<sup>ξ</sup><sup>p</sup> <sup>V</sup><sup>n</sup>þ1*=*<sup>2</sup> <sup>ξ</sup> *=*V<sup>n</sup> ξ

It is easy to show that, when using (5)–(7), the particles' total masses will remain

<sup>ξ</sup><sup>p</sup> þ e

unchanged, and the particles' total internal energies will be equal to the energy calculated for the given material as a whole, i.e. the following relationships hold:

<sup>ξ</sup><sup>p</sup> <sup>V</sup>ξp, enþ1*=*<sup>2</sup>

Major difficulties in implementing the particle method are associated with the Eulerian step, and namely, with calculations of mass and internal energy fluxes from cell to cell. In the PIC method, when a particle migrates to a neighbor cell, its mass and energy "migrate" with it. Because of the discrete (and, accordingly, nonmonotonic) character of the transfer of mass and all the quantities defined per unit mass, this method delivers highly non-monotonic quantity profiles. Section 4 provides a detailed description of the monotonization algorithm for the PIC method.

� �

nþ1*=*2 <sup>ξ</sup> � en ξ

� �

� �

<sup>ξ</sup> <sup>m</sup><sup>ξ</sup> <sup>¼</sup> <sup>X</sup>

p e nþ1*=*2

<sup>ξ</sup><sup>p</sup> þ ρ

p

u~nþ1*=*<sup>2</sup>

<sup>p</sup> � *<sup>y</sup>*<sup>n</sup> p

yp <sup>¼</sup> *<sup>λ</sup>*<sup>y</sup> � <sup>u</sup>~<sup>n</sup>þ1*=*<sup>2</sup>

ρn ξp*=*ρ<sup>n</sup>

<sup>p</sup> <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup> yp � τ*:*

<sup>p</sup> <sup>¼</sup> yn

yp � mp*:* (2)

<sup>τ</sup> , (3)

yp ; (4)

<sup>ξ</sup> , (5)

, (6)

*:* (7)

<sup>ξ</sup><sup>p</sup> mξ<sup>p</sup>*:* (8)

calculated:

*Recent Advances in Numerical Simulations*

were τ = tn+1- tn

formulas:

Ppx <sup>¼</sup> <sup>X</sup> p

> u~ <sup>n</sup>þ<sup>1</sup> <sup>2</sup> xp <sup>¼</sup> <sup>x</sup>~nþ1*=*<sup>2</sup>

.

u<sup>n</sup>þ1*=*<sup>2</sup>

ρ nþ1*=*2 <sup>ξ</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup><sup>n</sup>

ρ nþ1*=*2 <sup>ξ</sup> <sup>V</sup><sup>ξ</sup> <sup>¼</sup> <sup>X</sup>

**4. Eulerian step**

**46**

e nþ1*=*2 <sup>ξ</sup><sup>p</sup> <sup>¼</sup> en

> V<sup>n</sup>þ1*=*<sup>2</sup> <sup>ξ</sup><sup>p</sup> <sup>¼</sup> Vn

p ρ nþ1*=*2

xnþ1*=*<sup>2</sup> <sup>p</sup> <sup>¼</sup> xn u~nþ1*=*<sup>2</sup>

<sup>p</sup> � xn p <sup>τ</sup> , <sup>u</sup><sup>~</sup>

3.Coefficients *λ*<sup>x</sup> ¼ Pcx*=*Ppx, *λ*<sup>y</sup> ¼ Pcy*=*Ppy are calculated.

xp <sup>¼</sup> *<sup>λ</sup>*<sup>x</sup> � <sup>u</sup>~<sup>n</sup>þ1*=*<sup>2</sup>

**3.4 Determination of particle velocity, density and energy**

<sup>p</sup> <sup>þ</sup> <sup>u</sup><sup>n</sup>þ1*=*<sup>2</sup>

4.Remapping of particles density and energy to the cell as a whole.

These algorithms are listed in the order of their execution in the Eulerian step after the monotonization algorithm.
