Dear sir,

I have read 'A Time-Accurate Algorithm for Chemical

Non-Equiplibrium Vicous Flows at All Speeds" of 'J. S. Shuen et. al.

and I have some problems but I cannot contact the authors. Could you

give me some suggestions for the following questions please? --------------------------------------------------------------------

1. After doing some mathematical manipulations, we obtain the

Navier-Stokes equations in a conservation form in curvilinear

coordinates:

(P)[d(Q_hat)/d(tou)] + [d(Q_star)/d(t)] + [d(E_star)/d(sine)] +

[d(F_star)/d(eta)] = H

where Q_hat is J*[ p_g u v h]_transpose

tou is the pseudo time

t is the physical time

H is the source term

The author say that "the eigenvalues (in the sine-direction) in

the pseudo-time can be obtained from the matrix [(P_transpose)*A],

where A is the jacobian [d(E_star)/d(Q_hat)]."

My problem is what do I do with the eigenvalues obtained?

Actually I don't know why we have to find them. Use them to find the

delta_tou? How? and do I have to consider the eigenvalues in

eta-direction? What is the different influent on the pseudo-time term

of these two set of eigenvalues? -----------------------------------------------------------------------

2.After discretization, the dual time-stepping integration

method is chosen. (Actually I still have the problem how to apply this

method to this equation, the author don't say in deep detail.)

The author say that "the physical time step size, delta_t, is

determined based on the characteristic evolution of the unsteady flow

under consideration, the pseudo-time step size, delta_tou, is

determined based on the numerical stability of the algorithm and can be

adjusted to give the optimum convergence rate for the pseudo-time

iteration procedure."

My problem is I do not understand how to find delta_t and

delta_tou, and how to adjust it? Could you give me some explaination or

some examples please? ----------------------------------------------------------------------

3. The author said that he used a modified strongly implicit

procedure. Could you give me some explaination about this procedure

please or give me some examples of how to apply them please? ---------------------------------------------------------------------

Thank you very much sir.

Best regards,

Atit Koonsrisuk

The strong implicit procedure (SIP) is also called the Stone method in applied math. circles. It is basically a kind of incomplete cholesky algorithm. You can find it explained in the book by Ferziger and Peric.