Contractive difference schemes for symmetric hyperbolic systems
Authors:
Philip Brenner and Vidar Thomée
Journal:
Math. Comp. 25 (1971), 205217
MSC:
Primary 65N05
MathSciNet review:
0297151
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Consider the initialvalue problem for a constantcoefficient symmetric hyperbolic system with initialvalues vanishing in a halfspace. Consider also a finite difference operator consistent with the system. Conditions are given in terms of the orders of dissipation and accuracy which ensure that the solution of the discrete problem tends to zero exponentially with the meshwidth in halfspaces where the solution of the continuous problem vanishes.
 [1]
Mats
Y. T. Apelkrans, On difference schemes for hyperbolic
equations with discontinuous initial values, Math. Comp. 22 (1968), 525–539. MR 0233527
(38 #1848), http://dx.doi.org/10.1090/S00255718196802335276
 [2]
Philip
Brenner and Vidar
Thomée, Stability and convergence rates in
𝐿_{𝑝} for certain difference schemes, Math. Scand.
27 (1970), 5–23. MR 0278549
(43 #4279)
 [3]
Philip
Brenner and Vidar
Thomée, Estimates near discontinuities for some difference
schemes, Math. Scand. 28 (1971), 329–340
(1972). MR
0305613 (46 #4743)
 [4]
S. Gerschgorin, "Über die Abgrenzung der Eigenwerte einer Matrix," Izv. Akad. Nauk SSSR, v. 7, 1931, pp. 749754.
 [5]
G.
W. Hedstrom, Norms of powers of absolutely convergent Fourier
series, Michigan Math. J. 13 (1966), 393–416.
MR
0203340 (34 #3193)
 [6]
G.
W. Hedstrom, Norms of powers of absolutely convergent Fourier
series in several variables, Michigan Math. J. 14
(1967), 493–495. MR 0222549
(36 #5599)
 [7]
G.
W. Hedstrom, The rate of convergence of some difference
schemes, SIAM J. Numer. Anal. 5 (1968),
363–406. MR 0230489
(37 #6051)
 [8]
HeinzOtto
Kreiss, Über Matrizen die beschränkte Halbgruppen
erzeugen, Math. Scand. 7 (1959), 71–80
(German). MR
0110952 (22 #1820)
 [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]
HeinzOtto
Kreiss and Einar
Lundqvist, On difference approximations with
wrong boundary values, Math. Comp. 22 (1968), 1–12. MR 0228193
(37 #3777), http://dx.doi.org/10.1090/S0025571819680228193X
 [11]
P.
D. Lax, Differential equations, difference equations and matrix
theory, Comm. Pure Appl. Math. 11 (1958),
175–194. MR 0098110
(20 #4572)
 [12]
Stanley
Osher, On systems of difference equations
with wrong boundary conditions, Math. Comp.
23 (1969),
567–572. MR 0247785
(40 #1046), http://dx.doi.org/10.1090/S00255718196902477856
 [13]
Beresford
Parlett, Accuracy and dissipation in difference schemes, Comm.
Pure Appl. Math. 19 (1966), 111–123. MR 0196957
(33 #5141)
 [14]
Vidar
Thomée, Parabolic difference operators, Math. Scand.
19 (1966), 77–107. MR 0209693
(35 #590)
 [15]
Vidar
Thomée, Stability theory for partial difference
operators, SIAM Rev. 11 (1969), 152–195. MR 0250505
(40 #3739)
 [1]
 M. Y. T. Apelkrans, "On difference schemes for hyperbolic equations with discontinuous initial values," Math. Comp., v. 22, 1968, pp. 525539. MR 38 #1848. MR 0233527 (38:1848)
 [2]
 Ph. Brenner & V. Thomée, "Stability and convergence rates in for certain difference schemes," Math. Scand., v. 27, 1970, pp. 523. MR 0278549 (43:4279)
 [3]
 Ph. Brenner & V. Thomée, "Estimates near discontinuities for some difference schemes," Math. Scand. (To appear.) MR 0305613 (46:4743)
 [4]
 S. Gerschgorin, "Über die Abgrenzung der Eigenwerte einer Matrix," Izv. Akad. Nauk SSSR, v. 7, 1931, pp. 749754.
 [5]
 G. W. Hedstrom, "Norms of powers of absolutely convergent Fourier series," Michigan Math. J., v. 13, 1966, pp. 393416. MR 34 #3193. MR 0203340 (34:3193)
 [6]
 G. W. Hedstrom, "Norms of powers of absolutely convergent Fourier series in several variables," Michigan Math. J., v. 14, 1967, pp. 493495. MR 36 #5599. MR 0222549 (36:5599)
 [7]
 G. W. Hedstrom, "The rate of convergence of some difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 363406. MR 37 #6051. MR 0230489 (37:6051)
 [8]
 H. O. Kreiss, "Über Matrizen die beschränkte Halbgruppen erzeugen," Math. Scand., v. 7, 1959, pp. 7180. MR 0110952 (22:1820)
 [9]
 H. O. Kreiss, "On difference approximations of dissipative type for hyperbolic equations," Comm. Pure Appl. Math., v. 17, 1964, pp. 335353. MR 29 #4210. MR 0166937 (29:4210)
 [10]
 H. O. Kreiss & E. Lundqvist, "On difference approximations with wrong boundary values," Math. Comp., v. 22, 1968, pp. 112. MR 37 #3777. MR 0228193 (37:3777)
 [11]
 P. D. Lax, "Differential equations, difference equations and matrix theory," Comm. Pure Appl. Math., v. 11, 1958, pp. 175194. MR 20 #4572. MR 0098110 (20:4572)
 [12]
 S. Osher, "On systems of difference equations with wrong boundary conditions," Math. Comp., v. 23, 1969, pp. 567572. MR 0247785 (40:1046)
 [13]
 B. Parlett, "Accuracy and dissipation in difference schemes," Comm. Pure Appl. Math., v. 19, 1966, pp. 111123. MR 33 #5141. MR 0196957 (33:5141)
 [14]
 V. Thomée, "Parabolic difference operators," Math. Scand., v. 19, 1966, pp. 77107. MR 35 #590. MR 0209693 (35:590)
 [15]
 V. Thomée, "Stability theory for partial difference operators," SIAM Rev., v. 11, 1969, pp. 152195. MR 40 #3739. MR 0250505 (40:3739)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N05
Retrieve articles in all journals
with MSC:
65N05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197102971511
PII:
S 00255718(1971)02971511
Keywords:
Symmetric hyperbolic system,
initialvalue problem,
difference scheme,
contractive,
dissipative,
stability,
accuracy,
Fourier transform
Article copyright:
© Copyright 1971
American Mathematical Society
