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## 8.2.2 Pressure Method

Take the divergence (instead of rotation) of

and use

(with ) to obtain the basic equations

 (8.65) (8.66)

Two-dimensional case, explicit:

 (8.67) (8.68) (8.69)

Apply finite difference scheme.

Note: Divergence condition must stay intact in the course of the calculation.
- Let the pressure be localized at the centers of the Euler cells
- Velocity components and are localized at the right and upper box sides, respectively

- Now approximate the divergence of the velocity by
 (8.70)

or, in geographical'' notation,
 (8.71)

for vanishing divergence.

- Apply the Lax scheme to 8.67-8.68 (all terms on the r.h.s. are taken at time ):
 (8.72) (8.73)

- Insert the new velocity components in 8.71 to find
 (8.74)

with
 (8.75)

- Now solve the Poisson equation 8.69 to obtain the pressures.
Problem: After applying the Poisson solver the pressures contain small errors which cause the central values and deviate from zero. The simple ansatz
 (8.76)

for the pressures is therefore not usable. Rather, we write
 (8.77)

Stability: Again, the conditions are
 (8.78)

Next: 8.2.3 Free Surfaces: Marker-and-Cell Up: 8.2 Incompressible Flow with Previous: 8.2.1 Vorticity Method
Franz J. Vesely Oct 2005