doi: 10.3934/dcdsb.2022074
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On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

1. 

School of Applied Mathematical and Physical Sciences, National Technical University of Athens, GR-157 80 Zografou, Greece

2. 

Department of Mathematics and Applied Mathematics, Division of Applied Mathematics: Differential Equations and Numerical Analysis, University of Crete, GR-700 13 Voutes Campus, Heraklion, Crete, Greece

*Corresponding author: Georgios E. Zouraris

Dedicated to the memory of Professor Vassilios A. Dougalis.

Received  November 2021 Revised  March 2022 Early access April 2022

We consider an initial and Dirichlet boundary value problem for a logarithmic Schrödinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete $ L_t^{\infty}(L_x^2) $ norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrödinger equation is provided, without regularizing its nonlinear term.

Citation: Panagiotis Paraschis, Georgios E. Zouraris. On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022074
References:
[1]

G. D. AkrivisV. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.  doi: 10.1007/BF01385769.

[2]

A. H. Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electronic J. Differential Equations, 2016 (2016), 1-9. 

[3]

W. BaoR. CarlesC. Su and Q. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.

[4]

W. BaoR. CarlesC. Su and Q. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.

[5]

V. BarbuM. Röckner and D. Zhang, The stochastic logarithmic Schrödinger equation, J. Math. Pures Appl., 107 (2017), 123-149.  doi: 10.1016/j.matpur.2016.06.001.

[6]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[7]

H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E, 68 (2003), 036607, 6 pp. doi: 10.1103/PhysRevE.68.036607.

[8]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[9]

R. Carles and G. Ferriere, Logarithmic Schrödinger equation with quadratic potential, Nonlinearity, 34 (2021), 8283-8310.  doi: 10.1088/1361-6544/ac3144.

[10]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[11]

T. Cazenave and A. Haraux, Équations d' evolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.

[12]

B. Cheng and Z. Guo, Regularized splitting spectral method for space-fractional logarithmic Schrödinger equation, Appl. Numer. Math., 167 (2021), 330-355.  doi: 10.1016/j.apnum.2021.05.003.

[13]

P. D'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.

[14]

S. De MartinoM. FlanagaC. Godano and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Letts., 63 (2003), 472-475. 

[15]

R. van Geleuken and A. V. Martin, Numerical investigation of the logarithmic Schrödinger model of quantum decoherence, Phys. Rev. A, preprint, 105 (2022), 032210. doi: 10.1103/PhysRevA.105.032210.

[16]

P. GuerreroJ. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl., 11 (2010), 79-87.  doi: 10.1016/j.nonrwa.2008.10.017.

[17]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin Heidelberg, 1991. doi: 10.1007/978-3-662-09947-6.

[18]

M. Hayashi, A note on the nonlinear Schrödinger equation in a general domain, Nonlinear Anal., 173 (2018), 99-122.  doi: 10.1016/j.na.2018.03.017.

[19]

E. F. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A, 32 (1985), 1201-1204. 

[20]

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics 44, Springer-Verlag, 2003.

[21]

H. LiX. Zhao and Y. Hu, Numerical solution of the regularized logarithmic Schrödinger equation on unbounded domains, Appl. Numer. Math., 140 (2019), 91-103.  doi: 10.1016/j.apnum.2019.01.018.

[22]

P. Paraschis and G. Zouraris, Backward Euler finite difference approximations of a logarithmic heat equation over a 2D rectangular domain, preprint, 2021, hal-03220015.

[23]

R. Plato, Concise Numerical Mathematics, Graduate Texts in Mathematics Volume 57, American Mathematical Society, Providence, Rhode Island, 2003. doi: 10.1090/gsm/057.

[24]

T. C. Scott and J. Shertzer, Solution of the logarithmic Schrödinger equation with a Coulomb potential, Journal of Physics Communications, 2 (2018), 075014. 

[25]

J. Shertzer and T. C. Scott, Solution of the 3D logarithmic Schrödinger equation with a central potential, Journal of Physics Communications, 4 (2020), 065004. 

[26]

K. Yasue, Quantum mechanics of nonconservative systems, Annals Phys., 114 (1978), 479-496. 

show all references

References:
[1]

G. D. AkrivisV. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.  doi: 10.1007/BF01385769.

[2]

A. H. Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electronic J. Differential Equations, 2016 (2016), 1-9. 

[3]

W. BaoR. CarlesC. Su and Q. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.

[4]

W. BaoR. CarlesC. Su and Q. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.

[5]

V. BarbuM. Röckner and D. Zhang, The stochastic logarithmic Schrödinger equation, J. Math. Pures Appl., 107 (2017), 123-149.  doi: 10.1016/j.matpur.2016.06.001.

[6]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[7]

H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E, 68 (2003), 036607, 6 pp. doi: 10.1103/PhysRevE.68.036607.

[8]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[9]

R. Carles and G. Ferriere, Logarithmic Schrödinger equation with quadratic potential, Nonlinearity, 34 (2021), 8283-8310.  doi: 10.1088/1361-6544/ac3144.

[10]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[11]

T. Cazenave and A. Haraux, Équations d' evolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.

[12]

B. Cheng and Z. Guo, Regularized splitting spectral method for space-fractional logarithmic Schrödinger equation, Appl. Numer. Math., 167 (2021), 330-355.  doi: 10.1016/j.apnum.2021.05.003.

[13]

P. D'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.

[14]

S. De MartinoM. FlanagaC. Godano and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Letts., 63 (2003), 472-475. 

[15]

R. van Geleuken and A. V. Martin, Numerical investigation of the logarithmic Schrödinger model of quantum decoherence, Phys. Rev. A, preprint, 105 (2022), 032210. doi: 10.1103/PhysRevA.105.032210.

[16]

P. GuerreroJ. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl., 11 (2010), 79-87.  doi: 10.1016/j.nonrwa.2008.10.017.

[17]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin Heidelberg, 1991. doi: 10.1007/978-3-662-09947-6.

[18]

M. Hayashi, A note on the nonlinear Schrödinger equation in a general domain, Nonlinear Anal., 173 (2018), 99-122.  doi: 10.1016/j.na.2018.03.017.

[19]

E. F. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A, 32 (1985), 1201-1204. 

[20]

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics 44, Springer-Verlag, 2003.

[21]

H. LiX. Zhao and Y. Hu, Numerical solution of the regularized logarithmic Schrödinger equation on unbounded domains, Appl. Numer. Math., 140 (2019), 91-103.  doi: 10.1016/j.apnum.2019.01.018.

[22]

P. Paraschis and G. Zouraris, Backward Euler finite difference approximations of a logarithmic heat equation over a 2D rectangular domain, preprint, 2021, hal-03220015.

[23]

R. Plato, Concise Numerical Mathematics, Graduate Texts in Mathematics Volume 57, American Mathematical Society, Providence, Rhode Island, 2003. doi: 10.1090/gsm/057.

[24]

T. C. Scott and J. Shertzer, Solution of the logarithmic Schrödinger equation with a Coulomb potential, Journal of Physics Communications, 2 (2018), 075014. 

[25]

J. Shertzer and T. C. Scott, Solution of the 3D logarithmic Schrödinger equation with a central potential, Journal of Physics Communications, 4 (2020), 065004. 

[26]

K. Yasue, Quantum mechanics of nonconservative systems, Annals Phys., 114 (1978), 479-496. 

Figure 1.  Example 3: Snapshots of the absolute value of the numerical approximations for $ t = 0 $, $ t = 0.5 $, $ t = 1.0 $, $ t = 1.5 $
Figure 2.  Example 3: Snapshots of the real and imaginary part of the numerical approximations for $ t = 1.5 $
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2919(-2) 4.3590(-2)
40 5.9250(-3) 1.95 1.1840(-2) 1.88
80 1.5771(-3) 1.90 3.1454(-3) 1.91
160 4.0169(-4) 1.97 8.0380(-4) 1.96
320 1.0106(-4) 1.99 2.0219(-4) 1.99
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2941(-2) 4.3628(-2)
40 5.9307(-3) 1.95 1.1851(-2) 1.88
80 1.5785(-3) 1.90 3.1482(-3) 1.91
160 4.0205(-4) 1.97 8.0450(-4) 1.96
320 1.0115(-4) 1.99 2.0237(-4) 1.99
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2919(-2) 4.3590(-2)
40 5.9250(-3) 1.95 1.1840(-2) 1.88
80 1.5771(-3) 1.90 3.1454(-3) 1.91
160 4.0169(-4) 1.97 8.0380(-4) 1.96
320 1.0106(-4) 1.99 2.0219(-4) 1.99
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2941(-2) 4.3628(-2)
40 5.9307(-3) 1.95 1.1851(-2) 1.88
80 1.5785(-3) 1.90 3.1482(-3) 1.91
160 4.0205(-4) 1.97 8.0450(-4) 1.96
320 1.0115(-4) 1.99 2.0237(-4) 1.99
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2921(-2) 4.3591(-2)
40 5.9252(-3) 1.95 1.1840(-2) 1.88
80 1.5771(-3) 1.90 3.1455(-3) 1.91
160 4.0169(-4) 1.97 8.0380(-4) 1.96
320 1.0106(-4) 1.99 2.0219(-4) 1.99
($ {\varepsilon} $CNFD) method with $ {\varepsilon}=0 $
Example 1
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
20 2.2921(-2) 4.3591(-2)
40 5.9252(-3) 1.95 1.1840(-2) 1.88
80 1.5771(-3) 1.90 3.1455(-3) 1.91
160 4.0169(-4) 1.97 8.0380(-4) 1.96
320 1.0106(-4) 1.99 2.0219(-4) 1.99
($ ε $CNFD) method with $ ε=0 $
Example 2
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
40 3.9987(-2) 4.9215(-1)
80 9.7244(-3) 2.03 1.0367(-1) 2.24
160 2.4242(-3) 2.00 2.5763(-2) 2.00
320 6.0708(-4) 1.99 6.4455(-3) 1.99
($ ε $CNFD) method with $ ε=\tau^2 $
Example 2
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
40 4.0281(-2) 4.9741(-1)
80 9.9408(-3) 2.01 1.0538(-1) 2.23
160 2.4835(-3) 2.00 2.6368(-2) 1.99
320 6.2222(-4) 1.99 6.5854(-3) 2.00
($ ε $CNFD) method with $ ε=0 $
Example 2
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
40 3.9987(-2) 4.9215(-1)
80 9.7244(-3) 2.03 1.0367(-1) 2.24
160 2.4242(-3) 2.00 2.5763(-2) 2.00
320 6.0708(-4) 1.99 6.4455(-3) 1.99
($ ε $CNFD) method with $ ε=\tau^2 $
Example 2
$ \nu $ $ \mathsf{E}^0(\mathfrak{f}(\nu)) $ Rate $ \mathsf{E}^\infty(\mathfrak{f}(\nu)) $ Rate
40 4.0281(-2) 4.9741(-1)
80 9.9408(-3) 2.01 1.0538(-1) 2.23
160 2.4835(-3) 2.00 2.6368(-2) 1.99
320 6.2222(-4) 1.99 6.5854(-3) 2.00
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