Chapter 8. Chebyshev spectral methods

8.2. CHEBYSHEV DIFFERENTIATION MATRICES TREFETHEN 1994 269
8.2. Chebyshev di erentiation matrices
[Just a sketch]
From now on \Chebyshev points" means Chebyshev extreme points.
Multiplication by the rst-order Chebyshev di erentiation matrix DN transforms a vector of data at the Chebyshev points into approximate derivatives
at those points:
2 3 2 3
v0 w0 D N
6
4
.
v N
7
5 =
6
4
.
w N
As usual, the implicit de nition of D N is as follows:
CHEBYSHEV SPECTRAL DIFFERENTIATION BY POLYNOMIAL INTERPOLA-
TION.
(1) Interpolate v by a polynomial q(x)=q N (x)
(2) Di erentiate the interpolant atthe grid points x j :
7
5 :
w j =(D Nv) j = q 0 (x j): (8:2:1)
Higher-order di erentiation matrices are de ned analogously. From this
de nition it is easy to work out the entries of D N in special cases. For N =1:
For N =2:
x =
2
3
7
x =
6
4 1
5 D1 =
;1
2
6
4
1
3
7
0
7
5
;1
D2 =
2
6
4
2 1
6 2 ;
4
1 2
1
2 ; 1 3
7
5:
2
3 2 ;2 1 2
3
7
1
2 0 ; 1 2
; 1 2 2 ; 3 7
5
2
: