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Summation by parts, projections, and stability. I


Author: Pelle Olsson
Journal: Math. Comp. 64 (1995), 1035-1065, S23
MSC: Primary 65M06; Secondary 65M12
DOI: https://doi.org/10.1090/S0025-5718-1995-1297474-X
MathSciNet review: 1297474
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Abstract: We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By rearranging the analytic equations slightly, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield stability on nonsmooth domains in two space dimensions. Using the same procedure, one can prove stability in higher dimensions as well.


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  • [1] M. Carpenter and J. Otto, High-order cyclo-difference techniques: A new methodology for finite differences, Tech. report, ICASE, NASA Langley Research Center, Hampton, VA 23681-0001, 1993.
  • [2] D. Gottlieb, M. Carpenter, and S. Abarbanel, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes, Tech. Report 93-21, ICASE, NASA Langley Research Center, Hampton, VA 23681-0001, 1993. MR 1275021 (94m:65127)
  • [3] H.-O. Kreiss and J. Lorenz, Initial-boundary value problems and the Navier-Stokes equations, Pure and Appl. Math., vol. 136, Academic Press, San Diego, CA, 1989. MR 998379 (91a:35138)
  • [4] H.-O. Kreiss and G. Scherer, Finite element and finite difference methods for hyperbolic partial differential equations, Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press, New York, 1974, pp. 195-212. MR 0349031 (50:1525)
  • [5] -, On the existence of energy estimates for difference approximations for hyperbolic systems, Tech. Report, Dept. of Scientific Computing, Uppsala University, 1977.
  • [6] P. Olsson, Stable approximation of symmetric hyperbolic and parabolic equations in several space dimensions, Tech. Report 138, Dept. of Scientific Computing, Uppsala Univ., Uppsala, Sweden, December 1991.
  • [7] -, Summation by parts, projections, and stability. II, Math. Comp., to appear. MR 1308459 (96a:65131)
  • [8] G. Scherer, Numerical computations with energy estimates schemes, Tech. report, Dept. of Scientific Computing, Uppsala Univ., Uppsala, Sweden, April 1977. In PhD thesis by G. Scherer, 1977.
  • [9] B. Strand, Summation by parts for finite difference approximations for $ d/dx$, J. Comput. Phys. 110 (1994), 47-67. MR 1259900 (94k:65126)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1297474-X
Article copyright: © Copyright 1995 American Mathematical Society

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