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Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

Published online by Cambridge University Press:  03 June 2015

Yong Zhang
Yong Zhang*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China
*
*Corresponding author.Email:sunny5zhang@gmail.com
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Abstract

We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

Keywords

35Q5565M0665M1265M2281-08Schrodinger-Poisson systemCrank-Nicolson schemesemi-implicit schemecompact finite difference methodGronwall inequalitythe maximum principle
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013 , pp. 1357 - 1388
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Bao, W. and Cai, Y. Y., Optimal error estimate of finite difference methods for the Gross-Pitavskii equation with angular momentum rotation, Math. Comp.,to appear.Google Scholar
[2]Bao, W., Jin, S. and Markowich, P. A., Time-splitting spectral approximations for the Schrodinger equation in the semiclassical regime, J. Comput.Phys., 175 (2002), 487–524.Google Scholar
[3]Bao, W., Mauser, N. J. and Stimming, H. P., Effective one particle quantum dynamics of electrons: A numerical study of the Schrodinger-Poisson-Xa model, Comm. Math. Sci., 1 (2003), 809–831.Google Scholar
[4]Cazenave, T., Semilinear Schrodinger equations, (Courant Lecture Notes in Mathematics vol. 10), New York University, Courant Institute of Mathematical Sciences, AMS, 2003.Google Scholar
[5]Chang, Q., Guo, B. and Jiang, H., Finite Difference Method for Generalized Zakharov Equa-tions, Math. Comp., 64 (1995), 537–553.Google Scholar
[6]Chang, Q., Jia, E. and Sun, W., Difference Schemes for Solving the Generalized Nonlinear Schrodinger Equation, J. Comput. Phys., 148 (1999), 397–415.Google Scholar
[7]Chen, H. B. and Xu, D., A compact difference scheme for an evolution equation with a weakly singular kernel, Numer. Math. Theor. Meth. Appl., 5 (2012), 559–572.Google Scholar
[8]De Leo, M. and Rial, D., Well posedness and smoothing effect of Schrodinger-Poisson equation, J. Math. Phys., 48 (2007), 093509.CrossRefGoogle Scholar
[9]Dion, C. M. and Cances, E., Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), 046706.Google Scholar
[10]Dong, X. C., A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrodinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), 7917–7922.Google Scholar
[11]Galssey, R. T., Convergence of an Energy-Preserving Scheme for the Zakharov Equations in one Space Dimension, Math. Comp., 97 (1992), 83–102.Google Scholar
[12]Harrison, R., Moroz, I. M. and Tod, K. P., A numerical study of Schrodinger-Newton equations, Nonlinearity, 16 (2003), 101–122.Google Scholar
[13]Landes, R., On Galerkin’s method in the existence theory of quasilinear elliptic equations, J. Funct. Anal., 39 (1980), 123–148.Google Scholar
[14]Larsson, S. and Thomee, V., Partial differential equations with numerical methods, Springer, 2009.Google Scholar
[15]Masaki, S., Energy solution to Schrodinger-Poisson system in the two-dimensional whole space, SIAM J. Math. Anal., 43 (2011), 2719–2731.Google Scholar
[16]Ringhofer, C. and Soler, J., Discrete Schrodinger-Poisson systems preserving energy and mass, Appl. Math. Lett., 13 (2000), 27–32.Google Scholar
[17]Mohanty, R. K., Jain, M. K. andMishra, B. N., A novel numerical method of O(h_a4) for threedimensional non-linear triharmonic equations, Commun. Comput. Phys., 12 (2012), 14171433.CrossRefGoogle Scholar
[18]Sanz-Serna, J. M., Methods for the Numerical Solution of the Nonlinear Schroedinger Equation, Math. Comp., 43 (1984), 21–27.Google Scholar
[19]Sanyasiraju, Y. V. S. S. and Mishra, N., Exponential compact higher order scheme for nonlin-ear steady convection-diffusion equations, Commun. Comput.Phys., 9 (2011), 897–916.Google Scholar
[20]Sekhar, T. V. S., Raju, B. H. S. and Sanyasiraju, Y. V. S. S., Higher-order compact scheme for the incompressible Navier-Stokes equations in spherical geometry, Commun. Comput.Phys., 11 (2012), 99–113.Google Scholar
[21]Shen, J., Tang, T. and Wang, L. L., Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011.Google Scholar
[22]Soba, A., A finite element method solver for time-dependent and stationary Schrodinger equations with a generic potential, Commun. Comput. Phys., 5 (2009), 914–927.Google Scholar
[23]SteinrUck, H., The one-dimensional Wigner-Poisson problem and its relation to the Schrodinger-Poisson problem, SIAM J. Math. Anal., 22 (1991), 957–972.Google Scholar
[24]Stimming, H. P., The IVP for the Schrodinger-Poisson-Xa equation in one dimension, Math. Models Methods Appl. Sci., 15 (2005), 1169–1180.Google Scholar
[25]Feng, X.-F., Li, Z.-L. and Qiao, Z.-H., High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients, J. Comput. Math., 29 (2011), 324–340.Google Scholar
[26]Tan, I. H., Snider, G. L., Chang, L. D. and Hu, E. L., A self-consistent solution of Schrodinger-Poisson equations using a nonuniform mesh, J. Appl. Phys., 68 (1990), 4071–4076.Google Scholar
[27]Wang, T., Maximum norm error bound of a linearized difference scheme for a coupled non-linear Schroinger equations, J. Comput. Appl. Math., 235 (2011), 4237–4250.Google Scholar
[28]Xie, S. S., Li, G. X. and Yi, S., Compact finite difference schemes with high accuracyfor one-dimensional nonlinear Schrodinger equation, Comput.Meth. Appl. Mech. Eng., 198 (2009), 1052–1060.Google Scholar
[29]Zhang, J., Multigrid Method and Fourth-Order Compact Scheme for 2D Poisson Equation with Unequal Mesh-SizeDiscretization, J. Comput. Phys., 179 (2002), 170–179.Google Scholar
[30]Zhang, Y. and Dong, X. C., On the computation of ground state and dynamics of Schrodinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), 2660–2676.Google Scholar