Finite difference approximations for d/dx which satisfy a summation by parts rule, have been evaluated, using different types of norms, H, by implementation in the symbolic language Maple. In the simpler diagonal norm, H = diag(λ0, λ1, …, λ2γ-1, 1, …) with all λi, positive, difference operators accurate of order γ≤4 at the boundary and accurate of order 2γ in the interior have been evaluated. However, it was found that the difference operators form multi-parameter families of difference operators when γ≥3. In the general full norm, H = diag(Ĥ, I), with Ĥ ∈ ℜγ+1×γ+1 being SPD, and the identity matrix, difference operators accurate or order γ = 3, 5 at the boundary and accurate of order γ + 1 in the interior have been computed. As in the diagonal norm case we obtain a multiparameter family of operators when γ≥3. Finally, a three-parameter family of difference approximations with accuracy three at and near the boundary and with accuracy four in the interior have been computed using restricted full norms. Here, H = diag(Ĥ I), with Ĥ ∈ ℜγ+2×γ+2 being SPD, and Ĥ (:, 1)γ= Ke1 where K is a constant e1 is the vector with the first element being one and the rest zero. Regardless of which norm we use, the parameters can be determined such that the bandwidth of the difference operators are minimized. This is of interest when parallel computers are used, since the bandwidth determines the memory requirement and also the amount of computational work. © 1994 Academic Press, Inc. © 1994 Academic Press, Inc.
