On the stability analysis of boundary conditions for the wave equation by energy methods. I. The homogeneous case
Authors:
T. HaDuong and P. Joly
Journal:
Math. Comp. 62 (1994), 539563
MSC:
Primary 35L05; Secondary 35B35, 65N99
MathSciNet review:
1216259
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Abstract: We reconsider the stability theory of boundary conditions for the wave equation from the point of view of energy techniques. We study, for the case of the homogeneous halfspace, a large class of boundary conditions including the socalled absorbing conditions. We show that the results of strong stability in the sense of Kreiss, studied from the point of view of the modal analysis by Trefethen and Halpern, always correspond to the decay in time of a particular energy. This result leads to the derivation of new estimates for the solution of the associated mixed problem.
 [1]
Björn
Engquist and Andrew
Majda, Radiation boundary conditions for acoustic and elastic wave
calculations, Comm. Pure Appl. Math. 32 (1979),
no. 3, 314–358. MR 517938
(80e:76041), http://dx.doi.org/10.1002/cpa.3160320303
 [2]
K.
O. Friedrichs, Symmetric hyperbolic linear differential
equations, Comm. Pure Appl. Math. 7 (1954),
345–392. MR 0062932
(16,44c)
 [3]
Moshe
Goldberg and Eitan
Tadmor, Convenient stability criteria for
difference approximations of hyperbolic initialboundary value
problems, Math. Comp. 44
(1985), no. 170, 361–377. MR 777269
(86k:65078), http://dx.doi.org/10.1090/S00255718198507772697
 [4]
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
 [5]
Reuben
Hersh, Mixed problems in several variables, J. Math. Mech.
12 (1963), 317–334. MR 0147790
(26 #5304)
 [6]
T. Haduong and P. Joly, A generalized image principle for the wave equation with absorbing boundary conditions and applications to fourth order schemes, INRIA report n. 1306 (1990).
 [7]
T.
HaDuong and P.
Joly, Energy identities for the wave equation with high order
boundary conditions, Mathematical and numerical aspects of wave
propagation phenomena (Strasbourg, 1991) SIAM, Philadelphia, PA, 1991,
pp. 267–274. MR
1106000
 [8]
Robert
L. Higdon, Initialboundary value problems for linear hyperbolic
systems, SIAM Rev. 28 (1986), no. 2,
177–217. MR
839822 (88a:35138), http://dx.doi.org/10.1137/1028050
 [9]
Robert
L. Higdon, Absorbing boundary conditions for
difference approximations to the multidimensional wave equation,
Math. Comp. 47 (1986), no. 176, 437–459. MR 856696
(87m:65131), http://dx.doi.org/10.1090/S00255718198608566964
 [10]
HeinzOtto
Kreiss, Initial boundary value problems for hyperbolic
systems, Comm. Pure Appl. Math. 23 (1970),
277–298. MR 0437941
(55 #10862)
 [11]
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
 [12]
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)
 [13]
Daniel
Michelson, Stability theory of difference
approximations for multidimensional initialboundary value
problems, Math. Comp. 40
(1983), no. 161, 1–45. MR 679433
(84d:65068), http://dx.doi.org/10.1090/S00255718198306794332
 [14]
Lloyd
N. Trefethen, Instability of difference models for hyperbolic
initialboundary value problems, Comm. Pure Appl. Math.
37 (1984), no. 3, 329–367. MR 739924
(86f:65162), http://dx.doi.org/10.1002/cpa.3160370305
 [15]
Lloyd
N. Trefethen and Laurence
Halpern, Wellposedness of oneway wave
equations and absorbing boundary conditions, Math. Comp. 47 (1986), no. 176, 421–435. MR 856695
(88b:65148), http://dx.doi.org/10.1090/S00255718198608566952
 [16]
Calvin
H. Wilcox, Asymptotic wave functions and energy distributions in
strongly propagative anisotropic media, J. Math. Pures Appl. (9)
57 (1978), no. 3, 275–321. MR 513101
(80j:35082)
 [1]
 B. Engquist and A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math. 32 (1979), 313357. MR 517938 (80e:76041)
 [2]
 K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345392. MR 0062932 (16:44c)
 [3]
 M. Goldberg and E. Tadmor, Convenient stability criteria for difference approximations of hyperbolic initial boundary value problems, Math. Comp. 44 (1985), 361377. MR 777269 (86k:65078)
 [4]
 B. Gustafsson, H. O. Kreiss, and A. Sundström, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 25 (1972), 649686. MR 0341888 (49:6634)
 [5]
 R. Hersch, Mixed problems in several variables, J. Math. Mech. 12 (1963), 317334. MR 0147790 (26:5304)
 [6]
 T. Haduong and P. Joly, A generalized image principle for the wave equation with absorbing boundary conditions and applications to fourth order schemes, INRIA report n. 1306 (1990).
 [7]
 , Energy identities for the wave equation with high order boundary conditions, Proc. First Internat. Conf. on Wave Propagation (Strasbourg, 1991), SIAM, Philadelphia, PA, 1991. MR 1106000
 [8]
 R. L. Higdon, Initial boundary value problems for linear hyperbolic systems, SIAM Rev. 28 (1986), 177217. MR 839822 (88a:35138)
 [9]
 , Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp. 47 (1986), 437460. MR 856696 (87m:65131)
 [10]
 H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277298. MR 0437941 (55:10862)
 [11]
 J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
 [12]
 A. Majda and S. Osher, Initial boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), 607675. MR 0410107 (53:13857)
 [13]
 D. Michelson, Stability theory of difference approximations for multidimensional initial boundary value problems, Math. Comp. 40 (1983), 145. MR 679433 (84d:65068)
 [14]
 L. N. Trefethen, Instability of difference models for hyperbolic initial boundary value problems, Comm. Pure Appl. Math. 37 (1984), 329367. MR 739924 (86f:65162)
 [15]
 L. N. Trefethen and L. Halpern, Wellposedness of oneway wave equations and absorbing boundary conditions, Math. Comp. 47 (1986), 421435. MR 856695 (88b:65148)
 [16]
 C. Wilcox, Asymptotic wave functions and energy distributions in strongly propagative anisotropic media, J. Math. Pures Appl. 57 (1978), 275321. MR 513101 (80j:35082)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199412162592
PII:
S 00255718(1994)12162592
Keywords:
Absorbing boundary conditions,
wellposedness,
energy estimates
Article copyright:
© Copyright 1994
American Mathematical Society
