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

*(English)*Zbl 0778.65057Summary: The stability characteristics of various compact fourth- and sixth-order spatial operators are used to assess the theory of B. Gustafsson, H.-O. Kreiss and A. SundstrĂ¶m [(G-K-S). Math. Comput. 26, 649-686 (1972; Zbl 0293.65076)] 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 is then sharpened to include only those spatial discretizations that are asymptotically stable (bounded, left half-plane eigenvalues).

It is shown that 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 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 closure of various orders. The technique uses the semidiscrete 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.

It is shown that 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 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 closure of various orders. The technique uses the semidiscrete 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.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35L45 | Initial value problems for first-order hyperbolic systems |

##### Keywords:

high order finite-difference schemes; asymptotic stability; stability; semidiscrete; eigenvalue; sixth-order schemes; G-K-S stability
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\textit{M. H. Carpenter} et al., Appl. Numer. Math. 12, No. 1--3, 55--87 (1993; Zbl 0778.65057)

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##### References:

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