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Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II


Authors: Moshe Goldberg and Eitan Tadmor
Journal: Math. Comp. 36 (1981), 603-626
MSC: Primary 65M10
DOI: https://doi.org/10.1090/S0025-5718-1981-0606519-9
MathSciNet review: 606519
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Abstract: Convenient stability criteria are obtained for difference approximations to initial-boundary value problems associated with the hyperbolic system $ {{\mathbf{u}}_t} = A{{\mathbf{u}}_x} + B{\mathbf{u}} + {\mathbf{f}}$ in the quarter plane $ x \geqslant 0$, $ t \geqslant 0$. The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outflow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well-known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized.


References [Enhancements On Off] (What's this?)

  • [1] M. Goldberg, "On a boundary extrapolation theorem by Kreiss," Math. Comp., v. 31, 1977, pp. 469-477. MR 0443363 (56:1733)
  • [2] M. Goldberg & E. Tadmor, "Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. I," Math. Comp., v. 32, 1978, pp. 1097-1107. MR 501998 (80a:65196)
  • [3] B. Gustafsson, H.-O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649-686. MR 0341888 (49:6634)
  • [4] H.-O. Kreiss, "Difference approximations for hyperbolic differential equations," Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. of Maryland, 1965), Academic Press, New York, 1966, pp. 51-58. MR 0207223 (34:7039)
  • [5] H.-O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703-714. MR 0241010 (39:2355)
  • [6] H.-O. Kreiss & J. Oliger, Methods for the Approximate Solution of Time Dependent Problems, GARP Publication Series No. 10, 1973.
  • [7] P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [8] S. Osher, "Stability of parabolic difference approximations to certain mixed initial boundary value problems," Math. Comp., v. 26, 1972, pp. 13-39. MR 0298990 (45:8039)
  • [9] G. Sköllermo, How the Boundary Conditions Affect the Stability and Accuracy of Some Implicit Methods for Hyperbolic Equations, Report No. 62, Dept. of Comput. Sci., Uppsala Univ., Uppsala, Sweden, 1975.
  • [10] G. Sköllermo, Error Analysis for the Mixed Initial Boundary Value Problem for Hyperbolic Equations, Report No. 63, Dept. of Comput. Sci., Uppsala Univ., Uppsala, Sweden, 1975.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0606519-9
Article copyright: © Copyright 1981 American Mathematical Society

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