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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.

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  • $\begingroup$ I have a hard time parsing your math -- maybe you're missing an equals sign? $\endgroup$ – David Ketcheson Apr 26 '15 at 8:20
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    $\begingroup$ Assuming $L$ is linear, you just expand your solution in terms of the eigenvectors of your discretization of $L$. With finite difference methods and periodic boundary conditions, those are discrete Fourier modes. The evolution of the $a_j$ is then determined by the eigenvalues of the discretization. $\endgroup$ – David Ketcheson Apr 26 '15 at 8:23
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    $\begingroup$ The question is why you want to expand in terms of a polynomial basis, rather than in terms of the eigenvector basis of $L$? $\endgroup$ – Wolfgang Bangerth Apr 26 '15 at 16:51
  • $\begingroup$ @WolfgangBangerth I'm expanding on Polynomial basis becasue I'm using gPC technique to do uncertainty quantification for all sorts of input Noise. $\endgroup$ – Amir Sagiv Apr 28 '15 at 11:05
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It depends on how well resolving your mesh is. On some level of coarseness you will first observe $2h$ waves (smallest resolvable waves on your mesh), and soon after, your solution will blow-up. That means that in spectral representation of your solution, coefficients of higher modes start to grow due to aliasing and spectral blockage. You first observe that growth of higher coefficients when you see mentioned waves in solution. What you need to do, in that case, is to refine the mesh.

I think you might be interested in Chapter 11 of J.Boyd- "Fourier and Chebyshev Spectral Methods", Dover, 2001.

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