Stability theory of difference approximations for multidimensional initialboundary value problems
Author:
Daniel Michelson
Journal:
Math. Comp. 40 (1983), 145
MSC:
Primary 65M10
MathSciNet review:
679433
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A stability theory is developed for dissipative difference approximations to multidimensional initialboundary value problems. The original differential problem should be strictly hyperbolic and the difference problem consistent with the differential one. An algebra of pseudodifference operators is built and later used to prove the stability of the difference approximation with variable coefficients. In addition, stability of the Cauchy problem for weakly dissipative difference approximations with variable coefficients is proved.
 [1]
M. S. Agranovich, "Theorem of matrices depending on parameters and its applications to hyperbolic systems," Functional Anal. Appl., v. 6, 1972, pp. 8593. (Russian)
 [2]
L.
S. Frank, Spaces of network functions, Mat. Sb. (N.S.)
86 (128) (1971), 187–233 (Russian). MR 0290583
(44 #7763)
 [3]
L.
S. Frank, Algèbre des opérateurs aux
différences finies, Proceedings of the International Symposium
on Partial Differential Equations and the Geometry of Normed Linear Spaces
(Jerusalem, 1972), 1972, pp. 24–55 (1973) (French, with English
summary). MR
0394283 (52 #15086)
 [4]
I.
Gohberg and L.
Rodman, On spectral analysis of nonmonic matrix and operator
polynomials. I. Reduction to monic polynomials, Israel J. Math.
30 (1978), no. 12, 133–151. MR 508259
(80a:15010), http://dx.doi.org/10.1007/BF02760835
 [5]
Bertil
Gustafsson, HeinzOtto
Kreiss, and Arne
Sundström, Stability theory of difference
approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 0341888
(49 #6634), http://dx.doi.org/10.1090/S00255718197203418883
 [6]
Bertil
Gustafsson, The convergence rate for difference
approximations to mixed initial boundary value problems, Math. Comput. 29 (1975), 396–406. MR 0386296
(52 #7154), http://dx.doi.org/10.1090/S00255718197503862967
 [7]
HeinzOtto
Kreiss, Initial boundary value problems for hyperbolic
systems, Comm. Pure Appl. Math. 23 (1970),
277–298. MR 0437941
(55 #10862)
 [8]
HeinzOtto
Kreiss, Stability theory for difference
approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 0241010
(39 #2355), http://dx.doi.org/10.1090/S00255718196802410107
 [9]
HeinzOtto
Kreiss, On difference approximations of the dissipative type for
hyperbolic differential equations, Comm. Pure Appl. Math.
17 (1964), 335–353. MR 0166937
(29 #4210)
 [10]
H.O. Kreiss, Difference Approximations for Time Dependent Problems, AGARD Lecture Series No. 73, 1975.
 [11]
Andrew
Majda and Stanley
Osher, Initialboundary value problems for hyperbolic equations
with uniformly characteristic boundary, Comm. Pure Appl. Math.
28 (1975), no. 5, 607–675. MR 0410107
(53 #13857)
 [12]
Andrew
Majda, James
McDonough, and Stanley
Osher, The Fourier method for nonsmooth
initial data, Math. Comp.
32 (1978), no. 144, 1041–1081. MR 501995
(80a:65197), http://dx.doi.org/10.1090/S00255718197805019954
 [13]
D. Michelson, InitialBoundary Value Problems for Hyperbolic Equations and Their Difference Approximation With Uniformly Characteristic Boundary, Ph.D. thesis, Dept. of Math. Sci., TelAviv Univ., June 1980.
 [14]
Stanley
Osher, Stability of difference approximations
of dissipative type for mixed initialboundary value problems. I,
Math. Comp. 23 (1969), 335–340. MR 0246530
(39 #7834), http://dx.doi.org/10.1090/S00255718196902465308
 [15]
Beresford
Parlett, Accuracy and dissipation in difference schemes, Comm.
Pure Appl. Math. 19 (1966), 111–123. MR 0196957
(33 #5141)
 [16]
Jeffrey
Rauch, \cal𝐿₂ is a continuable initial condition for
Kreiss’ mixed problems, Comm. Pure Appl. Math.
25 (1972), 265–285. MR 0298232
(45 #7284)
 [17]
Leonard
Sarason, Hyperbolic and other symmetrizable systems in regions with
corners and edges, Indiana Univ. Math. J. 26 (1977),
no. 1, 1–39. MR 0442495
(56 #877)
 [1]
 M. S. Agranovich, "Theorem of matrices depending on parameters and its applications to hyperbolic systems," Functional Anal. Appl., v. 6, 1972, pp. 8593. (Russian)
 [2]
 L. S. Frank, "Spaces of net functions," Mat. Sb., v. 86(128), 1971. MR 0290583 (44:7763)
 [3]
 L. S. Frank, "Algèbre des opérateurs aux differences finies," Israel J. Math., v. 13, 1972, pp. 2455. MR 0394283 (52:15086)
 [4]
 I. Gohberg & L. Rodman, "On spectral analysis of nonmonic matrix and operator polynomials, I," Israel J. Math., v. 30, 1978, pp. 133151. MR 508259 (80a:15010)
 [5]
 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. 649686. MR 0341888 (49:6634)
 [6]
 B. Gustafsson, "The convergence rate for difference approximations to mixed initial boundary value problems," Math. Comp., v. 29, 1975, pp. 396406. MR 0386296 (52:7154)
 [7]
 H.O. Kreiss, "Initial boundary value problems for hyperbolic systems," Comm. Pure Appl. Math., v. 23, 1970, pp. 277298. MR 0437941 (55:10862)
 [8]
 H.O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 0241010 (39:2355)
 [9]
 H.O. Kreiss, "On difference approximations of the dissipative type for hyperbolic differential equations," Comm. Pure Appl. Math., v. 17, 1964, pp. 335353. MR 0166937 (29:4210)
 [10]
 H.O. Kreiss, Difference Approximations for Time Dependent Problems, AGARD Lecture Series No. 73, 1975.
 [11]
 A. Majda & S. Osher, "Initialboundary value problems for hyperbolic equations with uniformly characteristic boundary," Comm. Pure Appl. Math., v. 28, 1975, pp. 607675. MR 0410107 (53:13857)
 [12]
 A. Majda, J. McDonough & S. Osher, "The Fourier method for nonsmooth initial data," Math. Comp., v. 32, 1978, pp. 10411081. MR 501995 (80a:65197)
 [13]
 D. Michelson, InitialBoundary Value Problems for Hyperbolic Equations and Their Difference Approximation With Uniformly Characteristic Boundary, Ph.D. thesis, Dept. of Math. Sci., TelAviv Univ., June 1980.
 [14]
 S. Osher, "Stability of difference approximations of dissipative type for mixed initialboundary value problems. I," Math. Comp., v. 23, 1969, pp. 335340. MR 0246530 (39:7834)
 [15]
 B. N. Parlett, "Accuracy and dissipation in difference schemes," Comm. Pure Appl. Math., v. 19, 1966, pp. 111123. MR 0196957 (33:5141)
 [16]
 J. Rauch, " is a continuable initial condition for Kreiss' mixed problems," Comm. Pure Appl. Math., v. 25, 1972, pp. 265285. MR 0298232 (45:7284)
 [17]
 L. Sarason, "Hyperbolic and other symmetrizable systems in regions with corners and edges," Indiana Univ. Math. J., v. 26, No. 1, 1977, pp. 139. MR 0442495 (56:877)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M10
Retrieve articles in all journals
with MSC:
65M10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198306794332
PII:
S 00255718(1983)06794332
Keywords:
Stability theory,
initialboundary value problems,
difference approximations,
hyperbolic systems
Article copyright:
© Copyright 1983
American Mathematical Society
