@article{M2AN_1991__25_6_643_0,
author = {Akrivis, G. D. and Dougalis, V. A.},
title = {On a class of conservative, highly accurate Galerkin methods for the Schr\"odinger equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {25},
year = {1991},
pages = {643-670},
mrnumber = {1135988},
zbl = {0744.65085},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_1991__25_6_643_0}
}
Akrivis, G. D.; Dougalis, V. A. On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 25 (1991) pp. 643-670. http://gdmltest.u-ga.fr/item/M2AN_1991__25_6_643_0/
[1] and , « On a conservative, high-order accurate finite element scheme for the "parabolic" equation », in Computational Acoustics, D. Lee, A. Cakmak, R. Vichnevetsky eds., v. 1, 17-26, Elsevier-North Holland, Amsterdam, 1990. | MR 1095058
[2] , and , Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), 818-847. | MR 448947 | Zbl 0378.65061
[3] , , and , Conservative high order schemes for the generalized Korteweg-de Vries equation, to appear. | Zbl 0824.65095
[4] , Galerkin methods for approximation of solutions of second order partial differential equations of Schrödinger type, Ph. D. Thesis, University of Göteborg, 1980.
[5] , Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50-64. | MR 159424 | Zbl 0123.11701
[6] , The numerical analysis of ordinary differential equations ; Runge-Kutta methods and general linear methods, John Wiley, Chichester, 1987. | MR 878564 | Zbl 0616.65072
[7] , Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979), 75-82. | MR 525638 | Zbl 0431.65052
[8] and , On the discretization in time of semilinear parabolic equations with nonsmooth initial data, Math. Comp. 49 (1987), 359-377. | MR 906176 | Zbl 0632.65097
[9] and , Stability of Runge-Kutta methods for stiff nonlinear differential equations, North Holland, Amsterdam, 1984. | MR 774402 | Zbl 0571.65057
[10] , An alternating direction method for Schrödinger's equation, SIAM J. Numer. Anal. 14 (1977), 1028-1032. | MR 474853 | Zbl 0372.65042
[11] and , On some high order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp. 45 (1985), 329-345. | MR 804927 | Zbl 0609.65064
[12] and , On optimal high order in time approximations for the Korteweg-de Vries equation, Math. Comp. 55 (1990), 473-496. | MR 1035935 | Zbl 0725.65107
[13] and , Ocean acoustic propagation by finite difference methods, Comput. Math. Appl. 14 (1987) No. 5. | MR 916083 | Zbl 0637.76080
[14] , and eds., Computational acoustics : wave propagation, Proceedings of the 1st IMACS symposium on computational acoustics, New Haven, 6-8 August 1986, vols. 1, 2, North Holland, Amsterdam, 1988. | MR 937265 | Zbl 0684.00026
[15] and , Problèmes aux limites non homogènes et applications, vol. 2, Dunod, Paris, 1968. | MR 247244 | Zbl 0165.10801
[16] , Mixed approximations of evolution problems, Comput. Meths. Appl. Mech. Engrg. 24 (1980), 137-163. | MR 597041 | Zbl 0457.73049
[17] , Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp. 43 (1984), 21-27. | MR 744922 | Zbl 0555.65061
[18] and , Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Num. Anal. 6 (1986), 25-42. | MR 967679 | Zbl 0593.65087
[19] and eds., Computational ocean acoustics, Invited lectures from the workshop held at Yale University, 1-3 August 1984, Comput. Math. Appl. 11 (1985) Nos 7-8. | MR 809597
[20] , « The parabolic approximation method », in Wave propagation and underwater acoustics, J. B. Keller and J. S. Papadakis eds., 224-287, Lecture Notes in Physics v. 70, Springer-Verlag, Berlin-Heidelberg, 1977. | MR 475274
[21] « Convergence estimates for semi-discrete Galerkin methods for initial-value problems », in Numerische, insbesondere approximations-theoretische Behandlung von Funktionalgleichungen, R. Ansorge and W. Törnig eds., 243-262, Lecture Notes in Mathematics v. 333, Springer-Verlag, Berlin-Heidelberg, 1973. | MR 458948 | Zbl 0267.65069
[22] , Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics v. 1054, Springer-Verlag, Berlin-Heidelberg, 1984. | MR 744045 | Zbl 0528.65052
[23] , « A dissipative Galerkin method for the numerical solution of first order hyperbolic equations », in Mathematical aspects of fînite elements in partial differential equations, C. de Boor ed., 147-169, Academic Press, New York, 1974. | MR 658322 | Zbl 0346.65056
[24] , and , On fully discrete Galerkin methods of second-order temporal accuracy for the Nonlinear Schrödinger Equation, to appear in Numer. Math. | MR 1103752 | Zbl 0739.65096
[25] , and , On optimal-order error estimates for the Nonlinear Schrödinger Equation, to appear. | MR 1211396 | Zbl 0774.65091