This is something I've been trying to figure out for a long time, and all I have is vague numerical results. I'm trying to answer the following question analytically:
Suppose I have a time dependent system:
$u_t = L(x, u, u_x, u_{xx}) , u(t=0,x) = u_0 $
I approximate differences with some finite difference scheme, for instance $u_{xx}^j = \frac{1}{dx^2} (u^{j-1} +u^{j+1} -2u^{j} )$
And I approximate each time step with an implicit FD method, for instance Runge-Kutta.
The Question: Given an orthonormal polynomial base $\{ p_j (x) \}_{j=0}^{\infty } $, and given that $ u_0 $ is as smooth as necessary, what can we say about the spectral coefficients $u(x,t_f) = \sum\limits_j a_j p_j (x)$ for large $t_f$ ?
I think this is closely tied, if not determined by, how differentiable is $u(x,t)$ , but I'm not sure about that either.