zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. (English) Zbl 1221.65220
The initial and boundary value problem in a bounded domain for two-dimensional linear and nonlinear Schrödinger equations with nonlinearity in a real-valued function at the linear term are numerically solved. By using the alternating direction implicit method fourth-order in space and second-order in time compact finite difference schemes on uniform meshes are constructed. At each time step the schemes are reduced to one-dimensional scale tridiagonal symmetric systems of algebraic equations, which is the well-known advantage of the considered numerical method. Theoretical results on stability and error estimates are obtained. In addition, six numerical examples are solved in details to illustrate the applicability of the presented finite difference schemes and the numerical results are compared with the exact solutions of the selected problems.

65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations
Full Text: DOI
[1] Argyris, J.; Haase, M.: An engineer’s guide to solitons phenomena: application of the finite element method, Comput. methods appl. Mech. engrg. 61, 71-122 (1987) · Zbl 0624.76020 · doi:10.1016/0045-7825(87)90117-4
[2] Arnold, A.: Numerically absorbing boundary conditions for quantum evolution equations, VLSI des. 6, 313-319 (1998)
[3] Chang, Q.; Jia, E.; Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation, J. comput. Phys. 148, 397-415 (1999) · Zbl 0923.65059 · doi:10.1006/jcph.1998.6120
[4] Clavero, C.; Gracia, J. L.; Jorge, J. C.: A uniformly convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems, IMA J. Numer. anal. 26, 155-172 (2006) · Zbl 1118.65092 · doi:10.1093/imanum/dri029
[5] Clavero, C.; Jorge, J. C.; Lisbona, F.; Shishkin, G. I.: A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems, Appl. numer. Math. 27, 211-231 (1998) · Zbl 0929.65058 · doi:10.1016/S0168-9274(98)00014-2
[6] Clavero, C.; Jorge, J. C.; Lisbona, F.; Shishkin, G. I.: An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems, IMA J. Numer. anal. 20, 263-280 (2000) · Zbl 0962.65080 · doi:10.1093/imanum/20.2.263
[7] Davydov, A. S.: Solitons in molecular systems, (1985) · Zbl 0597.35001
[8] Dehghan, M.: Alternating direction implicit methods for two-dimensional diffusion with a nonlocal boundary condition, Int. J. Comput. math. 72, 349-366 (1999) · Zbl 0949.65085 · doi:10.1080/00207169908804858
[9] Dehghan, M.: A new ADI technique for two-dimensional parabolic equation with an integral condition, Comput. math. Appl. 43, 1477-1488 (2002) · Zbl 1001.65094 · doi:10.1016/S0898-1221(02)00113-X
[10] Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. Simulation 71, 16-30 (2006) · Zbl 1089.65085 · doi:10.1016/j.matcom.2005.10.001
[11] Dehghan, M.; Mirzaei, D.: Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, Internat. J. Numer. methods engrg. 76, 501-520 (2008) · Zbl 1195.81007 · doi:10.1002/nme.2338
[12] Dehghan, M.; Mirzaei, D.: The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Eng. anal. Bound. elem. 32, 747-756 (2008) · Zbl 1244.65139
[13] Dehghan, M.; Shokri, A.: A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. math. Appl. 54, 136-146 (2007) · Zbl 1126.65092 · doi:10.1016/j.camwa.2007.01.038
[14] Dehghan, M.; Taleei, A.: A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. phys. Comm. 181, 43-51 (2010) · Zbl 1206.65207 · doi:10.1016/j.cpc.2009.08.015
[15] Dehghan, M.; Taleei, A.: Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. methods partial differential equations 26, 979-992 (2010) · Zbl 1195.65137 · doi:10.1002/num.20468
[16] Delfour, M.; Fortin, M.; Payre, G.: Finite difference solution of a nonlinear Schrödinger equation, J. comput. Phys. 44, 277-288 (1981) · Zbl 0477.65086 · doi:10.1016/0021-9991(81)90052-8
[17] Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C.: Solitons and nonlinear wave equations, (1982) · Zbl 0496.35001
[18] Jr., J. Douglas: On the numerical integration of $\partial $2u$\partial x2+\partial $2u$\partial y2=\partial u\partial $t by implicit methods, J. soc. Ind. appl. Math. 3, 42-65 (1955) · Zbl 0067.35802 · doi:10.1137/0103004
[19] Jr., J. Douglas; Peaceman, D.: Numerical solution of two-dimensional heat flow problems, Aiche J. 1, 505-512 (1955)
[20] Jr., J. Douglas; Rachford, H.: On the numerical solution of heat conduction problems in two and three space variables, Trans. amer. Math. soc. 82, 421-439 (1960) · Zbl 0070.35401 · doi:10.2307/1993056
[21] Gardner, L. R. T.; Gardner, G. A.; Zaki, S. I.; El Sahrawi, Z.: B-spline finite element studies of the non-linear Schrödinger equation, Comput. methods appl. Mech. engrg. 108, 303-318 (1993) · Zbl 0842.65083 · doi:10.1016/0045-7825(93)90007-K
[22] Glassey, R. T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. math. Phys. 18, 1794-1797 (1977) · Zbl 0372.35009 · doi:10.1063/1.523491
[23] Hajj, F. Y.: Solution of the Schrödinger equation in two and three dimensions, J. phys. B: at. Mol. phys. 18, 1-11 (1985)
[24] Hasegawa, A.: Optical solitons in fibers, (1989)
[25] Herbst, B. M.; Morris, J. L.; Mitchell, A. R.: Numerical experience with the nonlinear Schrödinger equation, J. comput. Phys. 60, 282-305 (1985) · Zbl 0589.65084 · doi:10.1016/0021-9991(85)90008-7
[26] Huang, W.; Xu, C.; Chu, S. T.; Chaudhuri, S. K.: The finite difference vector beam propagation method, J. lightwave technol. 10, 295-304 (1992)
[27] Ixaru, L. Gr.: Operations on oscillatory functions, Comput. phys. Comm. 105, 1-9 (1997) · Zbl 0930.65150 · doi:10.1016/S0010-4655(97)00067-2
[28] Kalita, J. C.; Chhabra, P.; Kumar, S.: A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation, J. comput. Appl. math. 197, 141-149 (2006) · Zbl 1101.65096 · doi:10.1016/j.cam.2005.10.032
[29] Karakashian, O.; Makridakis, C.: A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. comp. 67, 479-499 (1998) · Zbl 0896.65068 · doi:10.1090/S0025-5718-98-00946-6
[30] Kim, S.; Lim, H.: High-order schemes for acoustic waveform simulation, Appl. numer. Math. 57, 402-414 (2007) · Zbl 1113.65087 · doi:10.1016/j.apnum.2006.05.003
[31] Levy, M.: Parabolic equation methods for electromagnetic wave propagation, (2000) · Zbl 0943.78001
[32] Lim, H.; Kim, S.; Jr., J. Douglas: Numerical method for viscous and non-viscous wave equations, Appl. numer. Math. 57, 194-212 (2007) · Zbl 1116.65104 · doi:10.1016/j.apnum.2006.02.004
[33] Mohebbi, A.; Dehghan, M.: The use of compact boundary value method for the solution of two-dimensional Schrödinger equation, J. comput. Appl. math. 225, 124-134 (2009) · Zbl 1159.65081 · doi:10.1016/j.cam.2008.07.008
[34] Pathria, D.; Morris, J. L.: Pseudo-spectral solution of nonlinear Schrödinger equations, J. comput. Phys. 87, 108-125 (1990) · Zbl 0691.65090 · doi:10.1016/0021-9991(90)90228-S
[35] Peaceman, D.; Rachford, H.: The numerical solution of parabolic and elliptic equations, J. soc. Ind. appl. Math. 3, 28-41 (1955) · Zbl 0067.35801 · doi:10.1137/0103003
[36] Sanz-Serna, J. M.: Methods for the numerical solution of the nonlinear Schrödinger equation, Math. comp. 43, 21-27 (1984) · Zbl 0555.65061 · doi:10.2307/2007397
[37] Subasi, M.: On the finite difference schemes for the numerical solution of two dimensional Schrödinger equation, Numer. methods partial differential equations 18, 752-758 (2002) · Zbl 1014.65077 · doi:10.1002/num.10029
[38] Sulem, C.; Sulem, P. L.: The nonlinear Schrödinger equation, self-focusing and wave collapse, (1999) · Zbl 0928.35157
[39] Sulem, P. L.; Sulem, C.; Patera, A.: Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equation, Comm. pure appl. Math. 37, 755-778 (1984) · Zbl 0543.65081 · doi:10.1002/cpa.3160370603
[40] Taha, T. R.; Ablowitz, M. J.: Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical nonlinear Schrödinger equation, J. comput. Phys. 55, 203-230 (1984) · Zbl 0541.65082 · doi:10.1016/0021-9991(84)90003-2
[41] Tappert, F. D.: The parabolic approximation method, Lecture notes in physics 70, 224-287 (1977)
[42] Tourigny, Y.; Morris, J. L.: An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schrödinger equation, J. comput. Phys. 76, 103-130 (1988) · Zbl 0641.65090 · doi:10.1016/0021-9991(88)90133-7
[43] Twizell, E. H.; Bratsos, A. G.; Newby, J. C.: A finite-difference method for solving the cubic Schrödinger equation, Math. comput. Simulation 43, 67-75 (1997) · Zbl 0886.65101 · doi:10.1016/S0378-4754(96)00056-0
[44] Wu, L.: Dufort-frankel-type methods for linear and nonlinear Schrödinger equations, SIAM J. Numer. anal. 33, 1526-1533 (1996) · Zbl 0860.65102 · doi:10.1137/S0036142994270636
[45] Xie, S.; Li, G.; Yi, S.: Compact finite difference schemes with high accuracy for one-dimensional nonlinear schröinger equation, Comput. methods appl. Mech. engrg. 198, 1052-1060 (2009) · Zbl 1229.81011 · doi:10.1016/j.cma.2008.11.011
[46] Xu, Y.; Shu, C. -W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. comput. Phys. 205, 72-97 (2005) · Zbl 1072.65130 · doi:10.1016/j.jcp.2004.11.001
[47] Yanenko, N. N.: The method of fractional steps, (1971) · Zbl 0209.47103
[48] Zakharov, V. E.; Synakh, V. S.: The nature of self-focusing singularity, Sov. phys. JETP 41, 465-468 (1975)
[49] Zhang, F.; Pérez-Grarciz, V. M.; Vázquez, L.: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. math. Comput. 71, 165-177 (1995) · Zbl 0832.65136 · doi:10.1016/0096-3003(94)00152-T