Spectral Methods Influence Matrix Formulation

In a fully spectral representation, each primary dependent variable, S = S x, y,z, t), is approximated through a triple series expansion in terms of its spectral coefficient, Ku = This involves a double Fourier series along the two periodic directions (streamwise and spanwise, using N, and Fourier modes, respectively) and Ny + 1 

Chebyshev orthogonal polynomial series along the nonperiodic (shear) direction  

The time integration of Eqs. (1.1)-(1.3), where the g tensor is given by Eq. (1.4) and the stress tensor x by Eq. (1.6), is performed by first formally integrating these equations with respect to time from t = to t = +1. Then, the following discretized 

We describe next two slightly different solution procedures used to find the solution at the time step n + 1 . 

The first scheme is a classical mixed semi-implicit/explicit scheme [51, 52]. According to this scheme, all linear terms in Eqs. (1.11)-(1.13) are treated implicitly and 

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