Stable and accurate boundary treatments for compact, high-order finite-difference schemes

Mark H. Carpenter, David Gottlieb, Saul Abarbanel

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75 Scopus citations


The stability characteristics of various compact fourth- and sixth-order spatial operators sre used to assess the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semidiscrete initial boundary value problem (IBVP). 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 (LHP) eigenvalues). It is shown that many of the higher-order schemes that are G-S-K stable are not asymptotically stable. A series of compact fourth- and sixth-order schemes is developed, all of which are asymptotically and G-K-S stable for the scalar case. A systematic technique is then presented for constructing stable and accurate boundary closures of various orders. The technique uses the semidescrete summation-by-parts energy norm to guarantee asymptotic and G-K-S stability of the resulting boundary closure. Various fourth-order explicit and implicit discretizations are presented, all of which satisfy the summation-by-parts energy norm.

Original languageEnglish
Pages (from-to)55-87
Number of pages33
JournalApplied Numerical Mathematics
Issue number1-3
StatePublished - May 1993
Externally publishedYes


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