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## 8.1.2 Particle-in-Cell Method (PIC)

Consider an ideal gas; assume adiabatic equation of state (fast flow or slow conduction of heat): constant in a flowing element Lagrangian time derivative . Therefore
 (8.17)

(continuity equation for ).

Flow equations:
 (8.18) (8.19) (8.20)

Strategy:
- Discretize the time:
- Assume a 2D (or 3D) Euler lattice with
- Represent the (variable) local density by a number of particles in each cell
- Each particle represents a fluid element (not a molecule!) and carries a vector of properties,
 (8.21)

The properties of the Eulerian (space fixed) cells are sums over the particles they contain:
 (8.22) (8.23) (8.24)

with
 (8.25)

- Now rewrite 8.19 as
 (8.26)

and first treat only the part :
 (8.27) (8.28)

with
- Update the particle properties and , thus:
 (8.29)

- Now treat the Lagrangian part of equation 8.26, by letting the fluid particles move with appropriate velocities: Defining
 (8.30)

compute the particle velocities as a weighted sum over the adjacent Eulerian cells:
 (8.31)

where the weights are the overlap areas of a square of side length centered around particle and the nearest Euler cells . (See the particle-mesh method of 6.5.2; see Fig. 6.9.)
- Update the positions
 (8.32)

to complete the time step.

Particle-in-cell method. Note that pressure gradients are evaluated using an Eulerian grid, while the transport of mass, momentum and energy is treated in continuous space.

Next: 8.1.3 Smoothed Particle Hydrodynamics Up: 8.1 Compressible Flow without Previous: 8.1.1 Explicit Eulerian Methods
Franz J. Vesely Oct 2005