Chebyshev-collocation

The Chebyshev-collocation method. This example problem uses the Chebyshev -collocation method to approximate u(x,t) in a domain x e [-1,1] as a solution to the PDE... 

The weighted residual expression for this Chebyshev-collocation will be... 

Wright K (1964) Chebyshev collocation methods for ordinary difleientieil equations. Comput 16 358-365... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

The possibly peculiar spacing of the collocation points, crowding close both at the electrode and at the outer diffusion limit, does not matter too much, and seems unnecessary. For example, using only five internal points (that is, five apart from zero and unity), they are placed at the values 0.047, 0.231, 0.5, 0.769, 0.953, a series that is symmetrical about the midway point at 0.500. This spacing has been circumvented by Yen and Chapman [580], using Chebyshev polynomials that open out towards the outer limit. Their work has apparently not been followed up. 

From equation [32] it is clear that x = -a is not a collocation point and an inconsistency with Reynolds equation is thus avoided. This additional constraint may be interpreted as requiring that pressure gradients in the neighbourhood of the inlet point x = -a are small and it should be noted that the use of a Chebyshev series expansion necessitates that pressure gradient at an end point is either zero or infinite. The constraint is therefore appropriate from both physical and mathematical standpoints. 

This is one of the variants of the FEMs. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi or Chebyshev polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their uneven spacing, crowding closer both at the electrode and (perhaps strangely) at the outer limit, and the fact that the outer limit is always unity. This is discussed below. 

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