A conservative finite element method for the KortewegdeVries equation
Author:
Ragnar Winther
Journal:
Math. Comp. 34 (1980), 2343
MSC:
Primary 65N30; Secondary 35Q20
MathSciNet review:
551289
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Abstract: A finite element method for the 1periodic Kortewegde Vries equation is analyzed. We consider first a semidiscrete method (i.e., discretization only in the space variable), and then we analyze some unconditionally stable fully discrete methods. In a special case, the fully discrete methods reduce to twelve point finite difference schemes (three time levels) which have second order accuracy both in the space and time variable.
 [1]
Shmuel
Agmon, Lectures on elliptic boundary value problems, Prepared
for publication by B. Frank Jones, Jr. with the assistance of George W.
Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co.,
Inc., Princeton, N.J.TorontoLondon, 1965. MR 0178246
(31 #2504)
 [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]
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)
 [4]
J.
H. Bramble and J.
E. Osborn, Rate of convergence estimates for
nonselfadjoint eigenvalue approximations, Math.
Comp. 27 (1973),
525–549. MR 0366029
(51 #2280), http://dx.doi.org/10.1090/S00255718197303660299
 [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]
F. TAPPERT, Numerical Solutions of the Kortewegde Vries Equation and Its Generalizations by the SplitStep Fourier Method, Lectures in Appl. Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1974, pp. 215216.
 [7]
A.
C. Vliegenthart, On finitedifference methods for the Kortewegde
Vries equation, J. Engrg. Math. 5 (1971),
137–155. MR 0363153
(50 #15591)
 [8]
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)
 [9]
G.
B. Whitham, Linear and nonlinear waves, WileyInterscience
[John Wiley & Sons], New YorkLondonSydney, 1974. Pure and Applied
Mathematics. MR
0483954 (58 #3905)
 [1]
 S. AGMON, Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 1965. MR 0178246 (31:2504)
 [2]
 M. E. ALEXANDER & J. L. MORRIS, "Galerkin methods applied to some model equations for nonlinear dispersive waves," J. Computational Phys. (To appear.) MR 530003 (80c:76006)
 [3]
 J. L. BONA & R. SMITH, "The initialvalue problem for the Kortewegde Vries equation," Philos. Trans. Roy. Soc. London Ser. A, v. 278, 1975, pp. 555604. MR 0385355 (52:6219)
 [4]
 J. H. BRAMBLE & J. E. OSBORN, "Rate of convergence estimates for nonselfadjoint eigenvalue approximations," Math. Comp., v. 27, 1973, pp. 525549. MR 0366029 (51:2280)
 [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]
 F. TAPPERT, Numerical Solutions of the Kortewegde Vries Equation and Its Generalizations by the SplitStep Fourier Method, Lectures in Appl. Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1974, pp. 215216.
 [7]
 A. C. VLIEGENTHART, "Finitedifference methods for the Kortewegde Vries equation," J. Engrg. Math., v. 5, 1971, pp. 137155. MR 0363153 (50:15591)
 [8]
 L. B. WAHLBIN, "A dissipative Galerkin method for the numerical solution of first order hyperbolic equations," Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, Ed.), Academic Press, New York, 1974, pp. 147169. MR 0658322 (58:31929)
 [9]
 G. B. WHITHAM, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005512895
PII:
S 00255718(1980)05512895
Article copyright:
© Copyright 1980
American Mathematical Society
