A second-order low-regularity integrator for the nonlinear Schrödinger equation
In this paper, we analyze a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the d dimensional torus T d . The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76, 2022 ). It is explicit and efficient to...
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| Veröffentlicht in: | Advances in continuous and discrete models 2022-03, Vol.2022 (1), Article 23 |
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| creator | Ostermann, Alexander Wu, Yifei Yao, Fangyan |
| description | In this paper, we analyze a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the
d
dimensional torus
T
d
. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76,
2022
). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in
H
γ
(
T
d
)
for initial data in
H
γ
+
2
(
T
d
)
for any
γ
>
d
/
2
. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986,
2019
).
The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work. |
| doi_str_mv | 10.1186/s13662-022-03695-8 |
| format | Article |
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d
dimensional torus
T
d
. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76,
2022
). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in
H
γ
(
T
d
)
for initial data in
H
γ
+
2
(
T
d
)
for any
γ
>
d
/
2
. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986,
2019
).
The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.</description><identifier>ISSN: 2731-4235</identifier><identifier>ISSN: 1687-1839</identifier><identifier>EISSN: 2731-4235</identifier><identifier>EISSN: 1687-1847</identifier><identifier>DOI: 10.1186/s13662-022-03695-8</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Accuracy ; Analysis ; Approximation ; Difference and Functional Equations ; Error analysis ; Functional Analysis ; Mathematics ; Mathematics and Statistics ; Methods ; Ordinary Differential Equations ; Partial Differential Equations ; Schrodinger equation ; Toruses</subject><ispartof>Advances in continuous and discrete models, 2022-03, Vol.2022 (1), Article 23</ispartof><rights>The Author(s) 2022</rights><rights>The Author(s) 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-8484028a38b5ddefc075efbe3410430b236e2748d24d12cdc552259dd102443e3</citedby><cites>FETCH-LOGICAL-c363t-8484028a38b5ddefc075efbe3410430b236e2748d24d12cdc552259dd102443e3</cites><orcidid>0000-0003-0194-2481</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1186/s13662-022-03695-8$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1186/s13662-022-03695-8$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,864,27924,27925,41120,41488,42189,42557,51319,51576</link.rule.ids></links><search><creatorcontrib>Ostermann, Alexander</creatorcontrib><creatorcontrib>Wu, Yifei</creatorcontrib><creatorcontrib>Yao, Fangyan</creatorcontrib><title>A second-order low-regularity integrator for the nonlinear Schrödinger equation</title><title>Advances in continuous and discrete models</title><addtitle>Adv Cont Discr Mod</addtitle><description>In this paper, we analyze a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the
d
dimensional torus
T
d
. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76,
2022
). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in
H
γ
(
T
d
)
for initial data in
H
γ
+
2
(
T
d
)
for any
γ
>
d
/
2
. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986,
2019
).
The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.</description><subject>Accuracy</subject><subject>Analysis</subject><subject>Approximation</subject><subject>Difference and Functional Equations</subject><subject>Error analysis</subject><subject>Functional Analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Methods</subject><subject>Ordinary Differential Equations</subject><subject>Partial Differential Equations</subject><subject>Schrodinger equation</subject><subject>Toruses</subject><issn>2731-4235</issn><issn>1687-1839</issn><issn>2731-4235</issn><issn>1687-1847</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp9kM1KAzEUhYMoWLQv4GrAdTS5N5PJLEvxDwoK6jpMJ5l2ypi0yQzSF_MFfDFTR9CVi8O9i_Odyz2EXHB2xbmS15GjlEAZJKEsc6qOyAQK5FQA5sd_9lMyjXHDGIMSsGBqQp5mWbS1d4b6YGzIOv9Og10NXRXafp-1rrerUPU-ZE1Sv7aZ865rna1C9lyvw-eHad0qgXY3VH3r3Tk5aaou2unPPCOvtzcv83u6eLx7mM8WtEaJPVVCCQaqQrXMjbFNzYrcNkuLgjOBbAkoLRRCGRCGQ23qPAfIS2M4AyHQ4hm5HHO3we8GG3u98UNw6aQGiQo5lJIlF4yuOvgYg230NrRvVdhrzvShPD2Wp1N5-rs8rRKEIxST-fDcb_Q_1Bf1JHJQ</recordid><startdate>20220312</startdate><enddate>20220312</enddate><creator>Ostermann, Alexander</creator><creator>Wu, Yifei</creator><creator>Yao, Fangyan</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0003-0194-2481</orcidid></search><sort><creationdate>20220312</creationdate><title>A second-order low-regularity integrator for the nonlinear Schrödinger equation</title><author>Ostermann, Alexander ; 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d
dimensional torus
T
d
. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76,
2022
). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in
H
γ
(
T
d
)
for initial data in
H
γ
+
2
(
T
d
)
for any
γ
>
d
/
2
. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986,
2019
).
The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1186/s13662-022-03695-8</doi><orcidid>https://orcid.org/0000-0003-0194-2481</orcidid><oa>free_for_read</oa></addata></record> |
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| source | DOAJ Directory of Open Access Journals; Springer Nature OA Free Journals; Alma/SFX Local Collection; SpringerLink Journals - AutoHoldings |
| subjects | Accuracy Analysis Approximation Difference and Functional Equations Error analysis Functional Analysis Mathematics Mathematics and Statistics Methods Ordinary Differential Equations Partial Differential Equations Schrodinger equation Toruses |
| title | A second-order low-regularity integrator for the nonlinear Schrödinger equation |
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