Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Authors:
Xavier Antoine, Christophe Besse and Vincent Mouysset

Translated by:

Journal:
Math. Comp. **73** (2004), 1779-1799

MSC (2000):
Primary 65M12, 35Q40, 58J40, 26A33, 58J47

DOI:
https://doi.org/10.1090/S0025-5718-04-01631-X

Published electronically:
January 23, 2004

MathSciNet review:
2059736

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain with artificial boundary conditions set on the arbitrarily shaped boundary of . These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

**1.**I. Alonso-Mallo, and N. Reguera,*Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations*, SIAM J. Numer. Anal. 40 (1) (2002), pp. 134-158. MR**2003f:65159****2.**X. Antoine,*Fast approximate computation of a time-harmonic scattered field using the On-Surface Radiation Condition method*, IMA J. Appl. Math. 66 (2001), pp. 83-110. MR**2001m:78033****3.**X. Antoine, and C. Besse,*Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrödinger equation*, J. Math. Pures Appl. 80 (7) (2001), pp. 701-738. MR**2003h:35213****4.**X. Antoine, and C. Besse,*Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation*, J. Comput. Phys. 181 (1) (2003), pp. 157-175.**5.**A. Arnold,*Numerically absorbing boundary conditions for quantum evolution equations*, VLSI Design 6 (1-4) (1998), pp. 313-319.**6.**A. Arnold,*Mathematical concepts of open quantum boundary conditions*, Transport Theory Statist. Phys., 30 (4-6) (2001), pp. 561-584.**7.**A. Arnold, and M. Erhardt,*Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics*, J. Comput. Phys. 145 (1998), pp. 611-638. MR**99g:76084****8.**A. Arnold, and M. Erhardt,*Discrete transparent boundary conditions for the Schrödinger equation*, Riv. Math. Univ. Parma 6 (4) (2001), pp. 57-108. MR**2003a:35161****9.**V.A. Baskakov and A.V. Popov,*Implementation of transparent boundaries for the numerical solution of the Schrödinger equation*, Wave Motion 14 (1991), pp. 123-128. MR**92g:78001****10.**C.-H. Bruneau, L. Di Menza, and T. Lerhner,*Numerical resolution of some nonlinear Schrödinger-like equations in plasmas*, Numer. Methods Partial Differential Equations (1999), pp. 672-696. MR**2000f:82099****11.**P. Deuflhard, and F. Schmidt,*Discrete transparent boundary conditions for the numerical solution of Fresnel's equation*, Comput. Math. Appl. 29 (9) (1995), pp. 53-76. MR**95k:65083****12.**L. Di Menza,*Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain*, Numer. Funct. Anal. Optim. 18 (7-8) (1997), pp. 759-775. MR**98h:65036****13.**T. Fevens, and H. Jiang,*Absorbing boundary conditions for the Schrödinger equation*, SIAM J. Sci. Comput. 21 (1) (1999), pp. 255-282. MR**2000h:65144****14.**T. Friese, F. Schmidt, and D. Yevick,*A comparison of transparent boundary conditions for the Fresnel equation*, J. Comput. Phys. 168 (2001), pp. 433-444. MR**2002b:78027****15.**R. Gorenflo, and F. Mainardi,*Fractional Calculus: Integral and Differential Equations of Fractional Order*, in Fractals and Fractional Calculus in Continuum Mechanics, Ed. A. Carpinteri and F. Mainardi, Springer-Verlag, Wien 1997. MR**99g:26015****16.**T. Ha Duong, and P. Joly,*A generalized image principle for the wave equation with absorbing boundary condition and application to fourth order schemes*, Numer. Methods Partial Differential Equations 10 (1994), (4), pp. 411-434. MR**95f:65164****17.**T. Hagstrom,*Radiation boundary conditions for the numerical simulation of waves*, Acta Numer. (1999), pp. 47-106. MR**2002c:35171****18.**P. Henrici,*Fast Fourier methods in computational complex analysis*, SIAM Rev. 21 (4) (1979), pp. 481-541. MR**80i:65031****19.**Y.V. Kopylov, A.V. Popov, and A.V. Vinogradov,*Application of the parabolic wave equation to X-ray diffraction optics*, Optics Comm. 118 (1995), pp. 619-636.**20.**J.-P. Kuska,*Absorbing boundary conditions for the Schrödinger equation on finite intervals*, Phys. Rev. B 46 (8) (1992), pp.5000-5003.**21.**M. Lévy,*Parabolic Equation Methods for Electromagnetic Wave Propagation*, IEE, 2000. MR**2003b:78001****22.**C. Lubich,*Discretized fractional calculus*, SIAM J. Math. Anal. 17 (3) (1986), pp. 704-719. MR**87f:26006****23.**C. Lubich, and A. Schädle,*Fast convolution for non-reflecting boundary conditions*, SIAM J. Sci. Comput. 24 (1) (2002), pp. 161-182. MR**2003h:44007****24.**B. Mayfield,*Non Local Boundary Conditions for the Schrödinger Equation*, Ph.D. Thesis, University of Rhodes Island, Providence, RI, 1989.**25.**A. Schädle,*Non-reflecting boundary conditions for the two-dimensional Schrödinger equation*, Wave Motion 35 (2002), pp. 181-188. MR**2002h:78010****26.**F. Schmidt,*Construction of discrete transparent boundary conditions for Schrödinger-type equations*, Surveys Math. Indust. 9 (2) (1999), pp. 87-100. MR**2000j:65080****27.**F. Schmidt, and D. Yevick,*Discrete transparent boundary conditions for Schrödinger-type equations*, J. Comput. Phys. 134 (1997), pp. 96-107. MR**98e:81028****28.**F.D. Tappert,*The parabolic approximation method*, in J.B. Keller and J.S. Papadakis, Ed., Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, Vol. 70, pp. 224-287, Springer, Berlin, 1977. MR**57:14891****29.**A.I. Zayed,*Handbook of Function and Generalized Function Transformation*, CRC Press, 1996. MR**97h:44001**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65M12,
35Q40,
58J40,
26A33,
58J47

Retrieve articles in all journals with MSC (2000): 65M12, 35Q40, 58J40, 26A33, 58J47

Additional Information

**Xavier Antoine**

Affiliation:
Laboratoire de Mathématiques pour l’Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France

Email:
antoine@mip.ups-tlse.fr

**Christophe Besse**

Affiliation:
Laboratoire de Mathématiques pour l’Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France

Email:
besse@mip.ups-tlse.fr

**Vincent Mouysset**

Affiliation:
Laboratoire de Mathématiques pour l’Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France

Email:
mouysset@mip.ups-tlse.fr

DOI:
https://doi.org/10.1090/S0025-5718-04-01631-X

Keywords:
Schr\"{o}dinger equation,
non-reflecting boundary condition,
stability,
semi-discrete Crank-Nicolson-type scheme,
finite-element methods

Received by editor(s):
November 7, 2002

Received by editor(s) in revised form:
April 7, 2003

Published electronically:
January 23, 2004

Article copyright:
© Copyright 2004
American Mathematical Society