August  2021, 41(8): 3837-3867. doi: 10.3934/dcds.2021019

On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $

1. 

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

2. 

Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea

Received  July 2019 Revised  December 2020 Published  August 2021 Early access  January 2021

In this paper, we establish a convergence result for the operator splitting scheme
$ Z_{\tau} $
introduced by Ignat [12], with initial data in
$ H^1 $
, for the nonlinear Schrödinger equation:
$ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x), $
where
$ p >0 $
,
$ \lambda \in \{-1,1\} $
and
$ (x,t) \in \mathbb{R}^d \times [0,\infty) $
. We prove the
$ L^2 $
convergence of order
$ \mathcal{O}(\tau^{1/2}) $
for the scheme with initial data in the space
$ H^1 (\mathbb{R}^d) $
for the energy-subcritical range of
$ p $
.
Citation: Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
References:
[1]

R. Altmann and A. Ostermann, Splitting methods for constrained diffusion-reaction systems, Comput. Math. Appl., 74 (2017), 962-976.  doi: 10.1016/j.camwa.2017.02.044.

[2]

C. BesseB. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.  doi: 10.1137/S0036142900381497.

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc/Courant Institute of Mathematical Sciences, Providence, RI/York New, NY, 2003. doi: 10.1090/cln/010.

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.

[5]

J. EilinghoffR. Schnaubelt and K. Schratz, Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation, J. Math. Anal. Appl., 442 (2016), 740-760.  doi: 10.1016/j.jmaa.2016.05.014.

[6]

E. FaouA. Ostermann and K. Schratz, Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., 35 (2015), 161-178.  doi: 10.1093/imanum/dru002.

[7]

L. Gauckler and C. Lubich, Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math., 10 (2010), 275-302.  doi: 10.1007/s10208-010-9063-3.

[8]

L. Gauckler and C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169.  doi: 10.1007/s10208-010-9059-z.

[9]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, 2004.

[10]

E. Hansen and A. Ostermann, High-order splitting schemes for semilinear evolution equations (English summary), Bit Numer. Math., 56 (2016), 1303-1316.  doi: 10.1007/s10543-016-0604-2.

[11]

L. I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591.  doi: 10.1142/S0218202507002029.

[12]

L. I. Ignat, A splitting method for the nonlinear Schrödinger equation, J. Differential Equations, 250 (2011), 3022-3046.  doi: 10.1016/j.jde.2011.01.028.

[13]

L. I. Ignat and E. Zuazua, A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386.  doi: 10.1016/j.crma.2005.07.018.

[14]

L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390.  doi: 10.1137/070683787.

[15]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[16]

M. KnöllerA. Ostermann and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data (English summary), SIAM J. Numer. Anal., 57 (2019), 1967-1986.  doi: 10.1137/18M1198375.

[17]

J. Lu and J. L. Marzuola, Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics, Commun. Math. Sci., 13 (2015), 1051-1074.  doi: 10.4310/CMS.2015.v13.n5.a1.

[18]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equation, Math. Comp., 77 (2008), 2141-2153.  doi: 10.1090/S0025-5718-08-02101-7.

[19]

A. Ostermann, F. Rousset and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., (2020), to appear.

[20]

A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.  doi: 10.1007/s10208-017-9352-1.

[21]

K. SchratzY. Wang and X. Zhao, Low-regularity integrators for nonlinear Dirac equations, Math. Comp., 90 (2021), 189-214.  doi: 10.1090/mcom/3557.

[22]

R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[23]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.

[24]

T. Tao, Nonlinear Dispersive Equations. Local And Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

[25]

M. Thalhammer, Higher-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.  doi: 10.1137/060674636.

[26]

M. ThalhammerM. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.  doi: 10.1016/j.jcp.2008.10.008.

show all references

References:
[1]

R. Altmann and A. Ostermann, Splitting methods for constrained diffusion-reaction systems, Comput. Math. Appl., 74 (2017), 962-976.  doi: 10.1016/j.camwa.2017.02.044.

[2]

C. BesseB. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.  doi: 10.1137/S0036142900381497.

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc/Courant Institute of Mathematical Sciences, Providence, RI/York New, NY, 2003. doi: 10.1090/cln/010.

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.

[5]

J. EilinghoffR. Schnaubelt and K. Schratz, Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation, J. Math. Anal. Appl., 442 (2016), 740-760.  doi: 10.1016/j.jmaa.2016.05.014.

[6]

E. FaouA. Ostermann and K. Schratz, Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., 35 (2015), 161-178.  doi: 10.1093/imanum/dru002.

[7]

L. Gauckler and C. Lubich, Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math., 10 (2010), 275-302.  doi: 10.1007/s10208-010-9063-3.

[8]

L. Gauckler and C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169.  doi: 10.1007/s10208-010-9059-z.

[9]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, 2004.

[10]

E. Hansen and A. Ostermann, High-order splitting schemes for semilinear evolution equations (English summary), Bit Numer. Math., 56 (2016), 1303-1316.  doi: 10.1007/s10543-016-0604-2.

[11]

L. I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591.  doi: 10.1142/S0218202507002029.

[12]

L. I. Ignat, A splitting method for the nonlinear Schrödinger equation, J. Differential Equations, 250 (2011), 3022-3046.  doi: 10.1016/j.jde.2011.01.028.

[13]

L. I. Ignat and E. Zuazua, A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386.  doi: 10.1016/j.crma.2005.07.018.

[14]

L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390.  doi: 10.1137/070683787.

[15]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[16]

M. KnöllerA. Ostermann and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data (English summary), SIAM J. Numer. Anal., 57 (2019), 1967-1986.  doi: 10.1137/18M1198375.

[17]

J. Lu and J. L. Marzuola, Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics, Commun. Math. Sci., 13 (2015), 1051-1074.  doi: 10.4310/CMS.2015.v13.n5.a1.

[18]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equation, Math. Comp., 77 (2008), 2141-2153.  doi: 10.1090/S0025-5718-08-02101-7.

[19]

A. Ostermann, F. Rousset and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., (2020), to appear.

[20]

A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.  doi: 10.1007/s10208-017-9352-1.

[21]

K. SchratzY. Wang and X. Zhao, Low-regularity integrators for nonlinear Dirac equations, Math. Comp., 90 (2021), 189-214.  doi: 10.1090/mcom/3557.

[22]

R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[23]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.

[24]

T. Tao, Nonlinear Dispersive Equations. Local And Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

[25]

M. Thalhammer, Higher-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.  doi: 10.1137/060674636.

[26]

M. ThalhammerM. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.  doi: 10.1016/j.jcp.2008.10.008.

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