about pseudo time
Dear sir,
I have read 'A TimeAccurate Algorithm for Chemical NonEquiplibrium 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 NavierStokes 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 sinedirection) in the pseudotime 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 etadirection? What is the different influent on the pseudotime term of these two set of eigenvalues?  2.After discretization, the dual timestepping 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 pseudotime 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 pseudotime 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 
Re: about pseudo time
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.

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