The stability of numerical boundary treatments for compact high-order finite-difference schemes

The stability characteristics of various compact fourth- and sixth-order spatial operators are assessed with the theory of Gustafsson, Kreiss, and Sundström (G-K-S) for the semidiscrete initial boundary value problem. These results are generalized to the fully discrete case with a recently developed theory of Kreiss and Wu. In all cases, favorable comparisons are obtained between G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable (bounded, left half-plane eigenvalues). Many of the higher order schemes that are G-K-S stable are not asymptotically stable. A series of compact fourth- and sixth-order schemes that are both asymptotically and G-K-S stable for the scalar case are then developed.