Since the not-yet-solved-for phi2 and phi3 values may appear in the
coefficients or sources of the phi1 equations, another solution
operation for phi1 must be conducted *after* phi2 and phi3 have
been solved.

Indeed *many* repetitions of all the solutions
operations are usually needed until the fields of phi1, phi2, phi3 etc
are sufficiently consistent with each other, and with the
equations governing them, for the solution to be regarded as having
converged.

There is one set of circumstances in which *'many'* becomes *'far too
many'*: it is that in which:

- the phi1 equation contains a term such as:
(phi2 - phi1) * linkco

- the phi2 equation contains a corresponding term:
(phi1 - phi2) * linkco

**and**the 'linking coefficient' linkco is so large that these terms greatly exceed all the others.

The purpose of the PEA is to prevent the slow-down of convergence which, without it, large linking coefficients inevitably produce.

Later, when the IMMERSOL model of radiative heat transfer was introduced into PHOENICS, slow convergence was encountered whenever the emissivity of the radiating medium was very high. The PEA was therefore applied to this situation also, the two linked variables being TEM1, the first-phase temperature, and T3, the radiative temperature.

It is also permissible, as is sometimes preferred in combustion simulations, to use H1, the first-phase enthalpy, in place of TEM1.

The user needs to take no action in order to activate the PEA for these situations. Its action is automatic.

Internally, the PEA is coded within PHOENICS so as to be able to handle an arbitrary number of linked variables; users wishing to avail themselves of the facility are asked to contact CHAM for advice, giving details of the variables in question, and especially of the linking terms.

It should be remarked that the linkage does not have to be linear; indeed it is not for IMMERSOL, for which the linking term equals

constant * (TEM1**4 - T3**4)

This 'best estimate' is that which would be arrived at if
phi2 were solved in a point-by-point manner, whereby
its neighbour-point values are presumed (for the moment) **not** to change.

The equation for phi2P for a particular computational cell P can be expressed as:

sum over all neighbours of: ( const2N * phi2N )

+ source2P

- const2P * phi2P

+ linkco * (phi1P - phi2P) = 0

Therefore, in the equation (1) for phi1P, it is possible to
**replace** the
slow-convergence-causing term:

linkco * (phi2P - phi1P)

by

sum over all neighbours of: ( const2N * phi2N )

+ source2P

- const2P * phi2P

This substitution, together with the corresponding one for the phi2P
equation when it is **its** turn to be solved, proves to be decisive:
the
linking coefficients may be arbitrarily large without slowing-down
convergence.

There is a small disadvantage: the coefficients of **both** variables
must be computed and stored before the solution of either variable can
begin. However, this would become severe only if many variables were to
be solved simultaneously. For two variables it is trivial.