I would like to know as to what is pseudo-spectral code. Please if possible also throw some light on Fourier modes. State some books on cfd (specifically by British authors) for a beginner. Thabnkyou well in advance.

Dearest Meena,

i am happy to respond on your enquiry. Hi, did you

go through the books online in this forum. I went

through it. I hope you can get some books. Go through it. will meet you afterwards.

Hi Meena,

There are different spectral methods (e.g. Galerkin, collocation method, etc..) and pseudo-spectral is just one of these. Usually the spectral methods actually transform the equations (Partial Differential Equations) into the spectral space (and obtain Ordinary Differential equations) and solve the equations there in the spectral space (if a Fourier method is used, you can call the spectral space, a Fourier space, etc..).

The pseudo-spectral method is the one in which the equations are actually solved in the Physical space rather than in the spectral space (i.e. the euqations are solved for the density, velocities rather than for the coefficient of the series of these variables). The picture is not that simple, since some method do actually solve the equations in the spectral space, however non linear terms (i.e. terms like density times velocity, or velocity square, etc...) cannot be solved simply in the spectral space. These terms involve the multiplication of two series (convolution) and one usually transforms them back to the physical space to obtain the coefficient of the expansion of the product of the two series (e.g. if f is expanded in a fourier series and a_n are the fourier coefficient, and the same for g expanded in b_n; then the to find the coefficient of the expansion of the product f*g, one uses a_n and b_n to obtain f and g in the physical space; then one multiplies f*g and transform in the spectral space to obtain the coefficient -say c_n - of the product a*b). These are also called pseudo-spectral.

A (fourier) mode is just the discrete index of the fourier coefficients. so n is the mode if a_n are the coefficients of the fourier expansion of a function f. Since n has positive and negative values, the mode usually stands for both. So if you have a_n with n=m and n=-m, then the amplitude of the mode m is just sqrt(a_m**2+a_-m**2) etc...

A good book that I know is

Canuto, Hussaini, Quarteroni and Zang, Spectral Methods in Fluid DYnamics, SPringer Verlag, Berlin 1988.

This book has a good practical approach, though it has also some chapter on the mathematical side.

If you do have specific questions, do not hesitate to post them here, I will do my best to help.

Cheers,

Patrick