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A conservative finite element method for the Korteweg-deVries equation

Author: Ragnar Winther
Journal: Math. Comp. 34 (1980), 23-43
MSC: Primary 65N30; Secondary 35Q20
MathSciNet review: 551289
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Abstract: A finite element method for the 1-periodic Korteweg-de Vries equation

$\displaystyle {u_t} + 2u{u_x} + {u_{xxx}} = 0$

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.

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  • [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 non-linear dispersive waves," J. Computational Phys. (To appear.) MR 530003 (80c:76006)
  • [3] J. L. BONA & R. SMITH, "The initial-value problem for the Korteweg-de Vries equation," Philos. Trans. Roy. Soc. London Ser. A, v. 278, 1975, pp. 555-604. 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. 525-549. 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. 373-404. MR 497916 (80i:35156)
  • [6] F. TAPPERT, Numerical Solutions of the Korteweg-de Vries Equation and Its Generalizations by the Split-Step Fourier Method, Lectures in Appl. Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1974, pp. 215-216.
  • [7] A. C. VLIEGENTHART, "Finite-difference methods for the Korteweg-de Vries equation," J. Engrg. Math., v. 5, 1971, pp. 137-155. 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. 147-169. MR 0658322 (58:31929)
  • [9] G. B. WHITHAM, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)

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Article copyright: © Copyright 1980 American Mathematical Society

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