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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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