I am simulating equations in spectral variables , the general form of equations is d/dt+2.v.k^2)E(k,t)=T(k,t), the function T(k,t) contains linear and non-linear terms. I have problems with stability, I would like to know which terms may have influence on it, the time step for exemple , or k_min, k_max, k_pic(initial spectrum),the caractristic time, ...

thanks in advance.

If you are using the EDQNM form of the transfer term, then you should have :

T(K) = int_{P,Q} (a_kpq E(P)E(Q) - b_kpq E(K)E(P)) dPdQ = T_nl - T_l

since the linear term T_l is negative, you can increase the stability of your scheme by writing it in an implicit form. This is possible if you are looking to a steady state solution - by forcing the small wave length for example.

If you are looking to a unsteady solution - turbulence decay for example - you should use a time scale at least as small as the smallest caracteristic time scale of your problem. In that case, I believe it is the turn over time of the smallest eddy - i.e. the turn over time associate to the Kolmogorov wave length K_kol. So if :

K_kol = (epsilon/nu^3)^{1/4}

then :

dt < 1/ ( 2 nu K_kol * K_kol )

where the dissipation rate epsilon is given by :

epsilon = int_{K} 2 nu K * K E(K) dK

hope that help