Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation
This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete H 1 framework to establish well-posedness and error estimates in the L ∞ norm. The nonlinearity f ( u ) of a NLS equation is assumed to satis...
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| Veröffentlicht in: | Japan journal of industrial and applied mathematics 2016-07, Vol.33 (2), p.427-470 |
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| creator | Saito, Norikazu Sasaki, Takiko |
| description | This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete
H
1
framework to establish well-posedness and error estimates in the
L
∞
norm. The nonlinearity
f
(
u
) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity
f
(
u
)
=
-
|
u
|
2
p
,
p
being a positive real number. Particularly, we offer the numerical blow-up time
T
(
h
,
τ
)
, where
h
and
τ
are discretization parameters of space and time variables. We prove that
T
(
h
,
τ
)
converges to the blow-up time
T
∞
of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of
T
(
h
,
τ
)
is at a second order rate in
τ
if the Crank–Nicolson scheme is applied to time discretization. |
| doi_str_mv | 10.1007/s13160-016-0218-8 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2063805834</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2063805834</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-77959be77acfc12096bb637c43d37f041e17d1680d7ec43413949ca6e90ae9c13</originalsourceid><addsrcrecordid>eNp1kM1KAzEURoMoWKsP4C7gevTeZpqfpRSrQsGFCu7CNJOxKdPJNJmh-mK-gC9m2hFcuQrcfOdevkPIJcI1AoibiAw5ZIA8gwnKTB6REUouM8XE2zEZgUo_AmB6Ss5iXAPkXCKOSD13jessLV1V2WAbY2nRtsF_uE3ROd_Qygfa-KZ2jS0CfTar8P1VuubdBmq3_SET6c51qz1XOzNQnafL2u-yvqXGb9q-O4zPyUlV1NFe_L5j8jq_e5k9ZIun-8fZ7SIzqUSXCaGmammFKExlcAKKL5ecCZOzkokKcrQoSuQSSmHTMEemcmUKbhUUVhlkY3I17E1Ftr2NnV77PjTppJ4AZxKmkuUphUPKBB9jsJVuQ2odPjWC3kvVg1SdpOq9VC0TMxmYmLJ7CX-b_4d-AJxcfKE</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2063805834</pqid></control><display><type>article</type><title>Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation</title><source>Springer Nature - Complete Springer Journals</source><creator>Saito, Norikazu ; Sasaki, Takiko</creator><creatorcontrib>Saito, Norikazu ; Sasaki, Takiko</creatorcontrib><description>This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete
H
1
framework to establish well-posedness and error estimates in the
L
∞
norm. The nonlinearity
f
(
u
) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity
f
(
u
)
=
-
|
u
|
2
p
,
p
being a positive real number. Particularly, we offer the numerical blow-up time
T
(
h
,
τ
)
, where
h
and
τ
are discretization parameters of space and time variables. We prove that
T
(
h
,
τ
)
converges to the blow-up time
T
∞
of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of
T
(
h
,
τ
)
is at a second order rate in
τ
if the Crank–Nicolson scheme is applied to time discretization.</description><identifier>ISSN: 0916-7005</identifier><identifier>EISSN: 1868-937X</identifier><identifier>DOI: 10.1007/s13160-016-0218-8</identifier><language>eng</language><publisher>Tokyo: Springer Japan</publisher><subject>Applications of Mathematics ; Computation ; Computational Mathematics and Numerical Analysis ; Convergence ; Discretization ; Finite difference method ; Mathematics ; Mathematics and Statistics ; Nonlinear equations ; Nonlinearity ; Original Paper ; Schrodinger equation ; Well posed problems</subject><ispartof>Japan journal of industrial and applied mathematics, 2016-07, Vol.33 (2), p.427-470</ispartof><rights>The JJIAM Publishing Committee and Springer Japan 2016</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-77959be77acfc12096bb637c43d37f041e17d1680d7ec43413949ca6e90ae9c13</citedby><cites>FETCH-LOGICAL-c316t-77959be77acfc12096bb637c43d37f041e17d1680d7ec43413949ca6e90ae9c13</cites><orcidid>0000-0002-2271-6355</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s13160-016-0218-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s13160-016-0218-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Saito, Norikazu</creatorcontrib><creatorcontrib>Sasaki, Takiko</creatorcontrib><title>Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation</title><title>Japan journal of industrial and applied mathematics</title><addtitle>Japan J. Indust. Appl. Math</addtitle><description>This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete
H
1
framework to establish well-posedness and error estimates in the
L
∞
norm. The nonlinearity
f
(
u
) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity
f
(
u
)
=
-
|
u
|
2
p
,
p
being a positive real number. Particularly, we offer the numerical blow-up time
T
(
h
,
τ
)
, where
h
and
τ
are discretization parameters of space and time variables. We prove that
T
(
h
,
τ
)
converges to the blow-up time
T
∞
of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of
T
(
h
,
τ
)
is at a second order rate in
τ
if the Crank–Nicolson scheme is applied to time discretization.</description><subject>Applications of Mathematics</subject><subject>Computation</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Convergence</subject><subject>Discretization</subject><subject>Finite difference method</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Nonlinear equations</subject><subject>Nonlinearity</subject><subject>Original Paper</subject><subject>Schrodinger equation</subject><subject>Well posed problems</subject><issn>0916-7005</issn><issn>1868-937X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kM1KAzEURoMoWKsP4C7gevTeZpqfpRSrQsGFCu7CNJOxKdPJNJmh-mK-gC9m2hFcuQrcfOdevkPIJcI1AoibiAw5ZIA8gwnKTB6REUouM8XE2zEZgUo_AmB6Ss5iXAPkXCKOSD13jessLV1V2WAbY2nRtsF_uE3ROd_Qygfa-KZ2jS0CfTar8P1VuubdBmq3_SET6c51qz1XOzNQnafL2u-yvqXGb9q-O4zPyUlV1NFe_L5j8jq_e5k9ZIun-8fZ7SIzqUSXCaGmammFKExlcAKKL5ecCZOzkokKcrQoSuQSSmHTMEemcmUKbhUUVhlkY3I17E1Ftr2NnV77PjTppJ4AZxKmkuUphUPKBB9jsJVuQ2odPjWC3kvVg1SdpOq9VC0TMxmYmLJ7CX-b_4d-AJxcfKE</recordid><startdate>20160701</startdate><enddate>20160701</enddate><creator>Saito, Norikazu</creator><creator>Sasaki, Takiko</creator><general>Springer Japan</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2271-6355</orcidid></search><sort><creationdate>20160701</creationdate><title>Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation</title><author>Saito, Norikazu ; Sasaki, Takiko</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-77959be77acfc12096bb637c43d37f041e17d1680d7ec43413949ca6e90ae9c13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Applications of Mathematics</topic><topic>Computation</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Convergence</topic><topic>Discretization</topic><topic>Finite difference method</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Nonlinear equations</topic><topic>Nonlinearity</topic><topic>Original Paper</topic><topic>Schrodinger equation</topic><topic>Well posed problems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Saito, Norikazu</creatorcontrib><creatorcontrib>Sasaki, Takiko</creatorcontrib><collection>CrossRef</collection><jtitle>Japan journal of industrial and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Saito, Norikazu</au><au>Sasaki, Takiko</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation</atitle><jtitle>Japan journal of industrial and applied mathematics</jtitle><stitle>Japan J. Indust. Appl. Math</stitle><date>2016-07-01</date><risdate>2016</risdate><volume>33</volume><issue>2</issue><spage>427</spage><epage>470</epage><pages>427-470</pages><issn>0916-7005</issn><eissn>1868-937X</eissn><abstract>This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete
H
1
framework to establish well-posedness and error estimates in the
L
∞
norm. The nonlinearity
f
(
u
) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity
f
(
u
)
=
-
|
u
|
2
p
,
p
being a positive real number. Particularly, we offer the numerical blow-up time
T
(
h
,
τ
)
, where
h
and
τ
are discretization parameters of space and time variables. We prove that
T
(
h
,
τ
)
converges to the blow-up time
T
∞
of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of
T
(
h
,
τ
)
is at a second order rate in
τ
if the Crank–Nicolson scheme is applied to time discretization.</abstract><cop>Tokyo</cop><pub>Springer Japan</pub><doi>10.1007/s13160-016-0218-8</doi><tpages>44</tpages><orcidid>https://orcid.org/0000-0002-2271-6355</orcidid></addata></record> |
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| ispartof | Japan journal of industrial and applied mathematics, 2016-07, Vol.33 (2), p.427-470 |
| issn | 0916-7005 1868-937X |
| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Applications of Mathematics Computation Computational Mathematics and Numerical Analysis Convergence Discretization Finite difference method Mathematics Mathematics and Statistics Nonlinear equations Nonlinearity Original Paper Schrodinger equation Well posed problems |
| title | Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation |
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