Cosine methods for nonlinear secondorder hyperbolic equations
Authors:
Laurence A. Bales and Vassilios A. Dougalis
Journal:
Math. Comp. 52 (1989), 299319, S15
MSC:
Primary 65M60
MathSciNet review:
955747
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We construct and analyze efficient, highorder accurate methods for approximating the smooth solutions of a class of nonlinear, secondorder hyperbolic equations. The methods are based on Galerkin type discretizations in space and on a class of fourthorder accurate twostep schemes in time generated by rational approximations to the cosine. Extrapolation from previous values in the coefficients of the nonlinear terms and use of preconditioned iterative techniques yield schemes whose implementation requires solving a number of linear systems at each time step with the same operator. optimalorder error estimates are proved.
 [1]
Garth
A. Baker, Vassilios
A. Dougalis, and Ohannes
Karakashian, On multistepGalerkin discretizations of semilinear
hyperbolic and parabolic equations, Nonlinear Anal. 4
(1980), no. 3, 579–597. MR 574375
(81j:65103), http://dx.doi.org/10.1016/0362546X(80)900942
 [2]
Laurence
A. Bales, Higherorder singlestep fully discrete approximations
for nonlinear secondorder hyperbolic equations, Comput. Math. Appl.
Part A 12 (1986), no. 45, 581–604. Hyperbolic
partial differential equations, III. MR 841989
(87h:65164)
 [3]
Laurence
A. Bales, Vassilios
A. Dougalis, and Steven
M. Serbin, Cosine methods for secondorder
hyperbolic equations with timedependent coefficients, Math. Comp. 45 (1985), no. 171, 65–89. MR 790645
(86j:65112), http://dx.doi.org/10.1090/S00255718198507906451
 [4]
James
H. Bramble and Peter
H. Sammon, Efficient higher order single step
methods for parabolic problems. I, Math.
Comp. 35 (1980), no. 151, 655–677. MR 572848
(81h:65110), http://dx.doi.org/10.1090/S0025571819800572848X
 [5]
J. H. Bramble & P. H. Sammon, "Efficient higher order single step methods for parabolic problems: Part II," unpublished manuscript.
 [6]
Chang
Ping Chen and Wolf
von Wahl, Das RandAnfangswertproblem für quasilineare
Wellengleichungen in Sobolevräumen niedriger Ordnung, J. Reine
Angew. Math. 337 (1982), 77–112 (German, with
English summary). MR 676043
(84b:35081), http://dx.doi.org/10.1515/crll.1982.337.77
 [7]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [8]
M.
Crouzeix and V.
Thomée, The stability in 𝐿_{𝑝}
and 𝑊¹_{𝑝} of the 𝐿₂projection onto
finite element function spaces, Math. Comp.
48 (1987), no. 178, 521–532. MR 878688
(88f:41016), http://dx.doi.org/10.1090/S00255718198708786882
 [9]
Constantine
M. Dafermos and William
J. Hrusa, Energy methods for quasilinear hyperbolic
initialboundary value problems. Applications to elastodynamics, Arch.
Rational Mech. Anal. 87 (1985), no. 3, 267–292.
MR 768069
(86k:35086), http://dx.doi.org/10.1007/BF00250727
 [10]
J.
E. Dendy Jr., An analysis of some Galerkin schemes for the solution
of nonlinear timedependent problems, SIAM J. Numer. Anal.
12 (1975), no. 4, 541–565. MR 0418477
(54 #6516)
 [11]
J.
E. Dendy Jr., Galerkin’s method for some highly nonlinear
problems, SIAM J. Numer. Anal. 14 (1977), no. 2,
327–347. MR 0433914
(55 #6884)
 [12]
Jim
Douglas Jr., Todd
Dupont, and Richard
E. Ewing, Incomplete iteration for timestepping a Galerkin method
for a quasilinear parabolic problem, SIAM J. Numer. Anal.
16 (1979), no. 3, 503–522. MR 530483
(80f:65117), http://dx.doi.org/10.1137/0716039
 [13]
Jim
Douglas Jr., Todd
Dupont, and Lars
Wahlbin, The stability in 𝐿^{𝑞} of the
𝐿²projection into finite element function spaces, Numer.
Math. 23 (1974/75), 193–197. MR 0383789
(52 #4669)
 [14]
Richard
E. Ewing, On efficient timestepping methods for nonlinear partial
differential equations, Comput. Math. Appl. 6 (1980),
no. 1 Issu, 1–13. MR 604081
(83g:65092), http://dx.doi.org/10.1016/08981221(80)900553
 [15]
Charles
I. Goldstein, Variational crimes and
𝐿^{∞} error estimates in the finite element method,
Math. Comp. 35 (1980), no. 152, 1131–1157. MR 583491
(81m:65169), http://dx.doi.org/10.1090/S00255718198005834910
 [16]
C.
I. Goldstein and L.
R. Scott, Optimal maximum norm error estimates for some finite
element methods for treating the Dirichlet problem, Calcolo
20 (1983), no. 1, 1–52. MR 747006
(85h:65239), http://dx.doi.org/10.1007/BF02575891
 [17]
Rolf
Rannacher and Ridgway
Scott, Some optimal error estimates for
piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661
(83e:65180), http://dx.doi.org/10.1090/S00255718198206456614
 [18]
Peter
Sammon, Fully discrete approximation methods for parabolic problems
with nonsmooth initial data, SIAM J. Numer. Anal. 20
(1983), no. 3, 437–470. MR 701091
(85a:65147), http://dx.doi.org/10.1137/0720031
 [19]
A.
H. Schatz, V.
C. Thomée, and L.
B. Wahlbin, Maximum norm stability and error estimates in parabolic
finite element equations, Comm. Pure Appl. Math. 33
(1980), no. 3, 265–304. MR 562737
(81g:65136), http://dx.doi.org/10.1002/cpa.3160330305
 [20]
A.
H. Schatz and L.
B. Wahlbin, On the quasioptimality in
𝐿_{∞} of the 𝐻¹projection into finite element
spaces, Math. Comp. 38
(1982), no. 157, 1–22. MR 637283
(82m:65106), http://dx.doi.org/10.1090/S00255718198206372836
 [1]
 G. A. Baker, V. A. Dougalis & O. A. Karakashian, "On multistep discretizations of semilinear hyperbolic and parabolic equations," Nonlinear Anal., v. 4, 1980, pp. 579597. MR 574375 (81j:65103)
 [2]
 L. A. Bales, "Higherorder singlestep fully discrete approximations for nonlinear secondorder hyperbolic equations," Comput. Math. Appl., v. 12A, 1986, pp. 581604. MR 841989 (87h:65164)
 [3]
 L. A. Bales, V. A. Dougalis & S. M. Serbin, "Cosine methods for secondorder hyperbolic equations with timedependent coefficients," Math. Comp., v. 45, 1985, pp. 6589. MR 790645 (86j:65112)
 [4]
 J. H. Bramble & P. H. Sammon, "Efficient higher order single step methods for parabolic problems: Part I," Math. Comp., v. 35, 1980, pp. 655677. MR 572848 (81h:65110)
 [5]
 J. H. Bramble & P. H. Sammon, "Efficient higher order single step methods for parabolic problems: Part II," unpublished manuscript.
 [6]
 V. C. Chen & W. v. Wahl, "Das RandAnfangswertproblem für quasilineare Wellengleichungen in Sobolevräumen niedriger Ordnung," J. Reine Angew. Math., v. 337, 1982, pp. 77112. MR 676043 (84b:35081)
 [7]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [8]
 M. Crouzeix & V. Thomée, "The stability in and of the projection onto finite element function spaces," Math. Comp., v. 48, 1987, pp. 521532. MR 878688 (88f:41016)
 [9]
 C. M. Dafermos & W. J. Hrusa, "Energy methods for quasilinear hyperbolic initialboundary value problems. Applications to elastodynamics," Arch. Rational Mech. Anal., v. 87, 1985, pp. 267292. MR 768069 (86k:35086)
 [10]
 J. E. Dendy, Jr., "An analysis of some Galerkin schemes for the solution of nonlinear timedependent problems," SIAM J. Numer. Anal., v. 12, 1975, pp. 541565. MR 0418477 (54:6516)
 [11]
 J. E. Dendy, Jr., "Galerkin's method for some highly nonlinear problems," SIAM J. Numer. Anal., v. 14, 1977, pp. 327347. MR 0433914 (55:6884)
 [12]
 J. Douglas, Jr., T. Dupont & R. E. Ewing, "Incomplete iteration for timestepping a Galerkin method for a quasilinear parabolic problem," SIAM J. Numer. Anal., v. 16, 1979, pp. 503522. MR 530483 (80f:65117)
 [13]
 J. Douglas, Jr., T. Dupont & L. Wahlbin, "The stability in of the projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193197. MR 0383789 (52:4669)
 [14]
 R. E. Ewing, "On efficient timestepping methods for nonlinear partial differential equations," Comput. Math. Appl., v. 6, 1980, pp. 113. MR 604081 (83g:65092)
 [15]
 C. I. Goldstein, "Variational crimes and error estimates in the finite element method," Math. Comp., v. 35, 1980, pp. 11311157. MR 583491 (81m:65169)
 [16]
 C. I. Goldstein & L. R. Scott, "Optimal maximum norm error estimates for some finite element methods for treating the Dirichlet problem," Calcolo, v. 20, 1984, pp. 152. MR 747006 (85h:65239)
 [17]
 R. Rannacher & R. Scott, "Some optimal error estimates for piecewise linear finite element approximations," Math. Comp., v. 38, 1982, pp. 437445. MR 645661 (83e:65180)
 [18]
 P. Sammon, "Fully discrete approximation methods for parabolic problems with nonsmooth initial data," SIAM J. Numer. Anal., v. 20, 1983, pp. 437470. MR 701091 (85a:65147)
 [19]
 A. H. Schatz, V. Thomée & L. B. Wahlbin, "Maximum norm stability and error estimates in parabolic finite element equations," Comm. Pure Appl. Math., v. 33, 1980, pp. 265304. MR 562737 (81g:65136)
 [20]
 A. H. Schatz & L. B. Wahlbin, "On the quasioptimality in of the projection into finite element spaces," Math. Comp., v. 38, 1982, pp. 122. MR 637283 (82m:65106)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M60
Retrieve articles in all journals
with MSC:
65M60
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909557479
PII:
S 00255718(1989)09557479
Article copyright:
© Copyright 1989
American Mathematical Society
