Understanding the Pseudospectral Method for Solving KdV Equation in MATLAB

[tiongwk]

Hi,

I'm having a problem coding pseudospectral method to solve KdV equation in matlab.
Can someone help me? thanks in advance.

KdV equation:
u_t+uu_x+u_xxx=0.

 

[matematikawan]

Just curious. Is Collocation Method also a pseudospectral method? I'm learning the wavelet collocation method to solve differential equations.

 

[tiongwk]

i don't think it is the same

 

[matematikawan]

Then the Collocation Method is a spectral method because it makes use of trial functions. Am I correct?

Really sorry to bother you all but I'm still learning numerical methods. From what I read, numerical methods for solving pde typically fall into one of the three classes, namely, finite difference /volume, finite element and spectral methods.

Where does the pseudospectral method fit in these classes?

 

[tiongwk]

pseudospectral is one of the spectral methods. pseudospectral often uses fast Fourier transform. I'm not so sure about the collocation method.

 

[matematikawan]

If I understood correctly, the spectral method is some sort like Fourier series method. Or do I?
Let say we want to solve a basic initial and boundary problem of linear heat conduction pde. Using the Fourier series method we assume separation of variables.
$$u(x,t)=\sum T_n(t)X_n(x)$$.

The spectral method numerical procedure truncates the Fourier series at N terms, u_N(x,t) and required its approximate satisfies the pde.
{ X_n } are prechosen functions which span a subspace.

If the approximate satisfies the equation at specified points, the method is called Collocation.
If the approximate satisfies the equation in projected sense, the method is called Galerkin.

What does the pseudospectral method satisfy ?