Abstract:Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1-periodic solutions of the Korteweg-de 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.
- Kanji Abe and Osamu Inoue, Fourier expansion solution of the Korteweg-de Vries equation, J. Comput. Phys. 34 (1980), no. 2, 202–210. MR 559996, DOI 10.1016/0021-9991(80)90105-9
- M. E. Alexander and J. Ll. Morris, Galerkin methods applied to some model equations for non-linear dispersive waves, J. Comput. Phys. 30 (1979), no. 3, 428–451. MR 530003, DOI 10.1016/0021-9991(79)90124-4
- Jerry Bona and Ridgway Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), no. 1, 87–99. MR 393887
- J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
- 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, DOI 10.1098/rsta.1978.0064
- I. S. Greig and J. Ll. Morris, A hopscotch method for the Korteweg-de-Vries equation, J. Comput. Phys. 20 (1976), no. 1, 64–80. MR 418475, DOI 10.1016/0021-9991(76)90102-9
- Peter D. Lax, Almost periodic solutions of the KdV equation, SIAM Rev. 18 (1976), no. 3, 351–375. MR 404889, DOI 10.1137/1018074
- Hans Schamel and Klaus Elsässer, The application of the spectral method to nonlinear wave propagation, J. Comput. Phys. 22 (1976), no. 4, 501–516. MR 449164, DOI 10.1016/0021-9991(76)90046-2 F. Tappert, "Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method," in Nonlinear Wave Motion (A. C. Newell, Ed.), Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, pp. 215-216.
- R. Temam, Sur un problème non linéaire, J. Math. Pures Appl. (9) 48 (1969), 159–172 (French). MR 261183
- Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
- 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 371088, DOI 10.1137/0711081
- A. C. Vliegenthart, On finite-difference methods for the Korteweg-de Vries equation, J. Engrg. Math. 5 (1971), 137–155. MR 363153, DOI 10.1007/BF01535405
- 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) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. MR 0658322
- Ragnar Winther, A conservative finite element method for the Korteweg-de Vries equation, Math. Comp. 34 (1980), no. 149, 23–43. MR 551289, DOI 10.1090/S0025-5718-1980-0551289-5 N. J. Zabusky, "Computation: Its role in mathematical physics innovation," J. Comput. Phys., v. 43, 1981, pp. 195-249. 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. 240-243.
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 40 (1983), 419-433
- MSC: Primary 65M60; Secondary 65M10
- DOI: https://doi.org/10.1090/S0025-5718-1983-0689464-4
- MathSciNet review: 689464