Convergence of Galerkin approximations for the Kortewegde Vries equation
Authors:
Garth A. Baker, Vassilios A. Dougalis and Ohannes A. Karakashian
Journal:
Math. Comp. 40 (1983), 419433
MSC:
Primary 65M60; Secondary 65M10
MathSciNet review:
689464
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1periodic solutions of the Kortewegde Vries equation are analyzed. Optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes. At each time level, the resulting system of nonlinear equations can be solved by Newton's method. It is shown that if a proper extrapolation is used as a starting value, then only one step of the Newton iteration is required.
 [1]
Kanji
Abe and Osamu
Inoue, Fourier expansion solution of the Kortewegde\thinspace
Vries equation, J. Comput. Phys. 34 (1980),
no. 2, 202–210. MR 559996
(81a:65113), http://dx.doi.org/10.1016/00219991(80)901059
 [2]
M.
E. Alexander and J.
Ll. Morris, Galerkin methods applied to some model equations for
nonlinear dispersive waves, J. Comput. Phys. 30
(1979), no. 3, 428–451. MR 530003
(80c:76006), http://dx.doi.org/10.1016/00219991(79)901244
 [3]
Jerry
Bona and Ridgway
Scott, Solutions of the Kortewegde Vries equation in fractional
order Sobolev spaces, Duke Math. J. 43 (1976),
no. 1, 87–99. MR 0393887
(52 #14694)
 [4]
J.
L. Bona and R.
Smith, The initialvalue problem for the Kortewegde Vries
equation, Philos. Trans. Roy. Soc. London Ser. A 278
(1975), no. 1287, 555–601. MR 0385355
(52 #6219)
 [5]
B.
Fornberg and G.
B. Whitham, A numerical and theoretical study of certain nonlinear
wave phenomena, Philos. Trans. Roy. Soc. London Ser. A
289 (1978), no. 1361, 373–404. MR 497916
(80i:35156), http://dx.doi.org/10.1098/rsta.1978.0064
 [6]
I.
S. Greig and J.
Ll. Morris, A hopscotch method for the KortewegdeVries
equation, J. Computational Phys. 20 (1976),
no. 1, 64–80. MR 0418475
(54 #6514)
 [7]
Peter
D. Lax, Almost periodic solutions of the KdV equation, SIAM
Rev. 18 (1976), no. 3, 351–375. MR 0404889
(53 #8688)
 [8]
Hans
Schamel and Klaus
Elsässer, The application of the spectral method to nonlinear
wave propagation, J. Computational Phys. 22 (1976),
no. 4, 501–516. MR 0449164
(56 #7469)
 [9]
F. Tappert, "Numerical solutions of the Kortewegde Vries equation and its generalizations by the splitstep Fourier method," in Nonlinear Wave Motion (A. C. Newell, Ed.), Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, pp. 215216.
 [10]
R.
Temam, Sur un problème non linéaire, J. Math.
Pures Appl. (9) 48 (1969), 159–172 (French). MR 0261183
(41 #5799)
 [11]
Roger
Temam, NavierStokes equations, Revised edition, Studies in
Mathematics and its Applications, vol. 2, NorthHolland Publishing
Co., AmsterdamNew York, 1979. Theory and numerical analysis; With an
appendix by F. Thomasset. MR 603444
(82b:35133)
 [12]
Vidar
Thomée and Burton
Wendroff, Convergence estimates for Galerkin methods for variable
coefficient initial value problems, SIAM J. Numer. Anal.
11 (1974), 1059–1068. MR 0371088
(51 #7309)
 [13]
A.
C. Vliegenthart, On finitedifference methods for the Kortewegde
Vries equation, J. Engrg. Math. 5 (1971),
137–155. MR 0363153
(50 #15591)
 [14]
Lars
B. Wahlbin, A dissipative Galerkin method for the numerical
solution of first order hyperbolic equations, Mathematical aspects of
finite elements in partial differential equations (Proc. Sympos., Math.
Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center,
Univ. of WisconsinMadison, Academic Press, New York, 1974,
pp. 147–169. Publication No. 33. MR 0658322
(58 #31929)
 [15]
Ragnar
Winther, A conservative finite element method
for the Kortewegde\thinspace Vries equation, Math. Comp. 34 (1980), no. 149, 23–43. MR 551289
(81a:65108), http://dx.doi.org/10.1090/S00255718198005512895
 [16]
N. J. Zabusky, "Computation: Its role in mathematical physics innovation," J. Comput. Phys., v. 43, 1981, pp. 195249.
 [17]
N. J. Zabusky & M. D. Kruskal, "Interaction of "solitons" in a collisionless plasma and the recurrence of initial states," Phys. Rev. Lett., v. 15, 1965, pp. 240243.
 [1]
 K. Abe & O. Inoue, "Fourier expansion solution of the Kortewegde Vries equation," J. Comput. Phys., v. 34, 1980, pp. 202210. MR 559996 (81a:65113)
 [2]
 M. E. Alexander & J. LL. Morris, "Galerkin methods applied to some model equations for nonlinear dispersive waves," J. Comput. Phys., v. 30, 1979, pp. 428451. MR 530003 (80c:76006)
 [3]
 J. Bona & R. Scott, "Solutions of the Kortewegde Vries equation in fractional order Sobolev spaces," Duke Math. J., v. 43, 1976, pp. 8799. MR 0393887 (52:14694)
 [4]
 J. Bona & R. Smith, "The initial value problem for the Kortewegde Vries equation," Philos. Trans. Roy. Soc. London Ser. A, v. 278, 1975, pp. 555604. MR 0385355 (52:6219)
 [5]
 B. Fornberg & G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Philos. Trans. Roy. Soc. London Ser. A, v. 289, 1978, pp. 373404. MR 497916 (80i:35156)
 [6]
 I. S. Greig & J. LL. Morris, "A Hopscotch method for the Kortewegde Vries equation," J. Comput. Phys., v. 20, 1976, pp. 6480. MR 0418475 (54:6514)
 [7]
 P. D. Lax, "Almost periodic solutions of the Kortewegde Vries equation," SIAM Rev., v. 18, 1976, pp. 351375. MR 0404889 (53:8688)
 [8]
 H. Schamel & K. Elsässer, "The application of the spectral method to nonlinear wave propagation," J. Comput. Phys., v. 22, 1976, pp. 501516. MR 0449164 (56:7469)
 [9]
 F. Tappert, "Numerical solutions of the Kortewegde Vries equation and its generalizations by the splitstep Fourier method," in Nonlinear Wave Motion (A. C. Newell, Ed.), Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, pp. 215216.
 [10]
 R. Temam, "Sur un problème non linéaire," J. Math. Pures Appl., v. 48, 1969, pp. 159172. MR 0261183 (41:5799)
 [11]
 R. Temam, NavierStokes Equations: Theory and Numerical Analysis, rev. ed., NorthHolland, Amsterdam, 1979. MR 603444 (82b:35133)
 [12]
 V. Thomée & B. Wendroff, "Convergence estimates for Galerkin methods for variable coefficient initial value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 10591068. MR 0371088 (51:7309)
 [13]
 A. C. Vliegenhart, "On finitedifference methods for the Kortewegde Vries equation," J. Engrg. Math., v. 5, 1971, pp. 137155. MR 0363153 (50:15591)
 [14]
 L. B. Wahlbin, "A dissipative Galerkin method for the numerical solution of first order hyperbolic equations," in Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, Ed.), Academic Press, New York, 1974, pp. 147169. MR 0658322 (58:31929)
 [15]
 R. Winther, "A conservative finite element method for the Kortewegde Vries equation," Math. Comp., v. 34, 1980, pp. 2343. MR 551289 (81a:65108)
 [16]
 N. J. Zabusky, "Computation: Its role in mathematical physics innovation," J. Comput. Phys., v. 43, 1981, pp. 195249.
 [17]
 N. J. Zabusky & M. D. Kruskal, "Interaction of "solitons" in a collisionless plasma and the recurrence of initial states," Phys. Rev. Lett., v. 15, 1965, pp. 240243.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M60,
65M10
Retrieve articles in all journals
with MSC:
65M60,
65M10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198306894644
PII:
S 00255718(1983)06894644
Article copyright:
© Copyright 1983
American Mathematical Society
