9.6 Spectral collocation differentiation

The equivalence (9.80View Equation) between the discrete truncated expansion and interpolation at Gauss-type points allows the approximation of the derivative of a function in a very simple way,
N -d- d-- d -d- ∑ -d- N dx f(x ) ≈ dx𝒫 N[f](x) = dx ℐ[f](x) = f(xj)dx ℓj (x). (9.81 ) j=0
Therefore, knowing the values of the function f at the collocation points, i.e., the Gauss-type points, we can construct its interpolant 𝒫d N, take an exact derivative thereof, and evaluate the result at the collocation points to obtain the values of the discrete derivative of f at these points. This leads to a matrix-vector multiplication, where the corresponding matrix elements Dij can be computed once and for all:
d ∑N ---f(xi) ≈ Dijf(xj), i = 0,1,...,N , (9.82 ) dx j=0
with
d Dij := ---ℓNj (xi). (9.83 ) dx
We give the explicit expressions for this differentiation matrix for Chebyshev polynomials both at Gauss and Gauss–Lobatto points (see, for example, [167, 237Jump To The Next Citation Point]).

Chebyshev–Gauss.

(| ---xi---- ||{ 2(1 − x2i) for i = j, Dij = || T ′ (x ) (9.84 ) |( -----N+1--′i------ for i ⁄= j, (xi − xj)T N+1(xj)
with a prime denoting differentiation.

Chebyshev–Gauss–Lobatto.

( | 2N 2 + 1 ||| − -------- for i = j = 0, |||| 6 ||| i+j ||| ci(−-1)---- for i ⁄= j, |{ cj(xi − xj ) Dij = (9.85 ) ||| ----xi--- |||| − 2 (1 − x2) for i = j = 1,...,(N − 1), ||| i ||| 2 ||( 2N--+-1- for i = j = N, 6
where c = 1 i for i = 1,...(N − 1 ) and c = 2 i for i = 0,N.
  Go to previous page Go up Go to next page