ABSTRACT
For each given even order of accuracy k, there is a stencil s = [s1; s2; :::; sk] that lls the \Toeplitz-type" structure on the interior rows of the desired matrixD, with bandwidth b = k. To modify the simplest matrix SD to obtain a higherorder approximation on the boundary, we only need to modify the upper left and lower right corners of SD. We dene A as a t-by-l matrix, A0 = PtAPl, with t = k and l = (3=2)k. Therefore, the general form of our desired hD = hD(A) will look like
h
0 0 A 0 0
0 0 0 0 s1 s2 sk 0 0 0 0 0 0 0 . . . . . . . . . 0 0 0 0 0 0 0 s1 s2 sk 0 0 0 0 0 0 A0 0 0
377777777777775 : (J.1)
The nonzero portions of A and A0 in hD(A) do not overlap, since we imposed the condition
N 2l 1 = 3k 1 = 3b 1: (J.2) Since A is a k-by-(3=2)k matrix, then for k = 4 we have A 2 R(46). Let ai = rowi(A), so that a1 = [a11 a12 a13 a14 a15 a16], and so on. The conditions on A will be described by a matrix equation. D(A) must
satisfy the row sum, column sum, and order constraints, and when we introduce a Vandermonde matrix Vi, the desired conditions on ai for k = 4 can be expressed by the four matrix equations:
V1a T 1 = V2a
T 2 = V3a
T 3 = V4a
T 4 = [0 1 0 0 0]
T ; (J.3)
which establish ve scalar constraints on each ai, i = 1; :::; 4. The rst constraint is that the row sum of ai is zero. The next four describe
the conditions on ai to obtain a fourth-order approximation.
