# zbMATH — the first resource for mathematics

Stable and accurate boundary treatments for compact, high-order finite- difference schemes. (English) Zbl 0778.65057
Summary: 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. Sundström [(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.

##### 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
Full Text:
##### References:
 [1] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, NASA contractor report 187628, (1991), ICASE Report No. 91-71, Hampton, VA [2] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite difference schemes solving hyperbolic systems: methodology and application to high order compact schemes, NASA contractor report 191436, (1993), ICASE Report No. 93-09, Hampton, VA [3] Gear, C., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 1145.65316 [4] Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. comp., 29, 130, 396-406, (1975) · Zbl 0313.65085 [5] Gustafsson, B.; Kreiss, H.-O.; Sundstrom, A., Stability theory of difference approximation for mixed initial boundary value problems II, Math. comp., 26, 649-686, (1972) · Zbl 0293.65076 [6] Kreiss, H.-O.; Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, () · Zbl 0355.65085 [7] H.-O. Kreiss and L. Wu, On the stability definition of difference approximations for the initial boundary value problem, Comm. Pure Appl. Math. (to appear). · Zbl 0782.65119 [8] Lele, S.K., Compact finite difference schemes with spectra-like resolution, Center for turbulence research manuscript, 107, (1990) [9] Reddy, S.C.; Trefethen, L.N., Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, (), (Proceedings of ICOSAHOM ’89, Como, Italy) · Zbl 0735.65070 [10] Strand, B., Summation by parts for finite difference approximations for $$ddx$$, (1991), Department of Scientific Computing, Uppsala University Uppsala, Sweden [11] Strikwerda, J.C., Initial boundary value problems for the method of lines, J. comput. phys., 34, 94-107, (1980) · Zbl 0441.65085 [12] Wylie, C.R., Advanced engineering mathematics, (1975), McGraw-Hill New York · Zbl 0313.00001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.