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**Chapter 3**

**Abstract**

**1. Introduction**

**43**

Particles

A Monotonic Method of Split

*Yury Yanilkin, Vladimir Shmelev and Vadim Kolobyanin*

The problem of correct calculation of the motion of a multicomponent (multimaterial) medium is the most serious problem for Lagrangian–Eulerian and Eulerian techniques, especially in multicomponent cells in the vicinity of interfaces.

There are two main approaches to solving the advection equation for a multicomponent medium. The first approach is based on the identification of interfaces and determining their position at each time step by the concentration field. In this case, the interface can be explicitly distinguished or reconstructed by the concentration field. The latter algorithm is the basis of widely used methods such as VOF. The second approach involves the use of the particle or marker method. In this case, the material fluxes of substances are determined by the particles with which certain masses of substances bind. Both approaches have their own advantages and drawbacks. The advantages of the particle method consist in the Lagrangian representation of particles and the possibility of" drawbacks. The main disadvantage of the particle method is the strong non-monotonicity of the solution caused by the discrete transfer of mass and mass-related quantities from cell to cell. This paper describes a particle method that is free of this drawback. Monotonization of the particle method is performed by spliting the particles so that the volume of matter flowing out of the cell corresponds to the volume calculated according to standard schemes of Lagrangian–Eulerian and Eulerian methods. In order not to generate an infinite chain of spliting, further split particles are reunited when certain conditions are met. The method is developed for modeling 2D and 3D gas-dynamic flows with accompanying processes, in which it is necessary to

preserve the history of the process at Lagrangian points.

**Keywords:** Eulerian method, PIC method, numerical simulation, gas-dynamic

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

**Chapter 3**
