Symmetrizable finite difference operators

Wade, Bruce A.

doi:10.1090/S0025-5718-1990-1011447-9

Symmetrizable finite difference operators

Author: Bruce A. Wade
Journal: Math. Comp. 54 (1990), 525-543
MSC: Primary 65M10
DOI: https://doi.org/10.1090/S0025-5718-1990-1011447-9
MathSciNet review: 1011447
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notion of a symmetrizable finite difference operator and prove that such operators are stable. We then present some sufficient conditions for symmetrizability. One of these extends H.-O. Kreiss' theorem on dissipative difference schemes for hyperbolic equations to a more general case with full (x, t)-variable coefficients.

[1] K. P. Bube and J. C. Strikwerda, Interior regularity estimates for elliptic systems of difference equations, SIAM J. Numer. Anal. 20 (1983), no. 4, 653–670. MR 708449, https://doi.org/10.1137/0720044
[2] Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
[3] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
[4] Heinz-Otto Kreiss, On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math. 17 (1964), 335–353. MR 166937, https://doi.org/10.1002/cpa.3160170306
[5] Heinz-Otto Kreiss, Über sachgemässe Cauchyprobleme, Math. Scand. 13 (1963), 109–128 (German). MR 168921, https://doi.org/10.7146/math.scand.a-10694
[6] P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gȧrding’s inequality, Comm. Pure Appl. Math. 19 (1966), 473–492. MR 206534, https://doi.org/10.1002/cpa.3160190409
[7] Daniel Michelson, Stability theory of difference approximations for multidimensional initial-boundary value problems, Math. Comp. 40 (1983), no. 161, 1–45. MR 679433, https://doi.org/10.1090/S0025-5718-1983-0679433-2
[8] John J. H. Miller, On power-bounded operators and operators satisfying a resolvent condition, Numer. Math. 10 (1967), 389–396. MR 220080, https://doi.org/10.1007/BF02162872
[9] Beresford Parlett, Accuracy and dissipation in difference schemes, Comm. Pure Appl. Math. 19 (1966), 111–123. MR 196957, https://doi.org/10.1002/cpa.3160190109
[10] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
[11] Hisayoshi Shintani and Kenji Tomoeda, Stability of difference schemes for nonsymmetric linear hyperbolic systems with variable coefficients, Hiroshima Math. J. 7 (1977), no. 1, 309–378. MR 448930
[12] John C. Strikwerda and Bruce A. Wade, An extension of the Kreiss matrix theorem, SIAM J. Numer. Anal. 25 (1988), no. 6, 1272–1278. MR 972453, https://doi.org/10.1137/0725071
[13] Eitan Tadmor, The equivalence of 𝐿₂-stability, the resolvent condition, and strict 𝐻-stability, Linear Algebra Appl. 41 (1981), 151–159. MR 649723, https://doi.org/10.1016/0024-3795(81)90095-1
[14] M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, N. J., 1984.
[15] Robert Vichnevetsky and John B. Bowles, Fourier analysis of numerical approximations of hyperbolic equations, SIAM Studies in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1982. With a foreword by Garrett Birkhoff. MR 675265
[16] B. A. Wade, Stability and sharp convergence estimates for symmetrizable difference operators, Ph. D. dissertation, University of Wisconsin-Madison, 1987.
[17] Olof B. Widlund, On the stability of parabolic difference schemes, Math. Comp. 19 (1965), 1–13. MR 170479, https://doi.org/10.1090/S0025-5718-1965-0170479-9