magnitudes of eigenvalues VS stiffness
Dear sir, I read from one numerical fluid book that "The condition and degree of stiffness of the system can be related to the relative magnitudes of the eigenvalues. When the eigenvalues differ greatly in magnitudes, convergence to a steady-state solution is usually slow. This occurs because greatly varying signal speeds appear in the equations and the traditional solution schemes attempt to honor all of them.". Actually I know how to find out the eigenvalues of the system, but I never know the physical meaning of the eigenvalues. Could anyone who know please explain this concept to me? I know that they can be used to find out the solution, but I don't know the physical relation between them and the solution. In the quoted sentences, the author say about signal speeds, what does signal mean? Every flow property? Please explain to me. Thank you very much.
Re: magnitudes of eigenvalues VS stiffness
I am no mathematician but I think I can give you some idea about the relationship between the magnitudes of the eigenvalues and stiffness. If you have a system of coupled partial differential equations of the form, U_t + A U_x = 0 where U is a vector (eg. p, u, T for a one-dimensional flow) and A is not a diagonal matrix.
Then A can be diagonalized (under some constraints) by a similarity transformation D = T^-1 A T, where T is the transformation matrix and D is the resulting diagonal matrix. Now actually if D is chosen to the eigenvalues of A and T is chosen as the matrix made up of the eigenvectors of A then the similarity transformation works.
Applying this transformation the system becomes decoupled with each individual equation being of the form w_t + d w_x = 0; this is like a simple plane wave equation with d as the wave speed. w is called a characteristic variable. So now the time step chosen is a function of 'd' and hence if the d's are quite different in magnitude one has to use the highest one to get the solution thus making the system stiff!!!
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