Summation by Parts for Finite Difference Approximations for d/dx
Abstract
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(λK; _{0}, λ _{1}, ..., λ _{2τ  1}, 1, ...) with all λ _{i} positive, difference operators accurate of order τ⩽4 at the boundary and accurate of order 2τ in the interior have been evaluated. However, it was found that the difference operators form multiparameter families of difference operators when τ⩾3. In the general full norm, H = diag(H∼, I), with H∼ ∈ R^{τ + 1 × τ + 1} being SPD, and I the identity matrix, difference operators accurate or order τ = 3, 5 at the boundary and accurate of order τ + 1 in the interior have been computed. As in the diagonal norm case we obtain a multiparameter family of operators when τ⩾3. Finally, a threeparameter 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(H∼ I), with H∼ ∈ R^{τ + 2 × τ + 2} being SPD, and H∼ (:, 1) = H∼ (1, :)τ = Ke_{1} , where K is a constant e_{1} 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.
 Publication:

Journal of Computational Physics
 Pub Date:
 January 1994
 DOI:
 10.1006/jcph.1994.1005
 Bibcode:
 1994JCoPh.110...47S