{"id":{"repo_id":"nus","oai_identifier":"oai:scholarbank.nus.edu.sg:10635/174472"},"canonical_url":"https://search.dev.ndltd.org/etd/nus/oai:scholarbank.nus.edu.sg:10635/174472","repository":{"repo_id":"nus","name":"National University of Singapore","base_url":"https://scholarbank.nus.edu.sg/oai/request"},"display":{"title":"ERROR ESTIMATES OF NUMERICAL METHODS FOR THE LONG-TIME DYNAMICS OF THE NONLINEAR KLEIN-GORDON EQUATION","abstract":"This work is devoted to the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\\varepsilon^2$ with $\\varepsilon \\in (0, 1]$ a dimensionless parameter.The analytical results indicate that the life-span of a smooth solution to this NKGE is at least up to the time at $O(\\varepsilon^{-2})$. Different numerical methods are adapted to discretize the problem and rigorous error bounds are established for the long-time dynamics.The numerical methods studied in this work include the finite difference methods, exponential wave integrator methods as well as the time-splitting methods and particular attentions are paid on the error bounds of different numerical methods up to the time $t= T_0/\\varepsilon^{\\beta}$ with $0 \\leq \\beta \\leq 2$ and $T_0$ fixed. As a by-product, the results are extended to solve an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\\varepsilon^{\\beta})$ in time. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.","abstract_html":"This work is devoted to the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by <span class=\"etd-inline-math\">\\varepsilon<sup>2</sup></span> with $\\varepsilon \\in (0, 1]$ a dimensionless parameter.The analytical results indicate that the life-span of a smooth solution to this NKGE is at least up to the time at <span class=\"etd-inline-math\">O(\\varepsilon<sup>-2</sup>)</span>. Different numerical methods are adapted to discretize the problem and rigorous error bounds are established for the long-time dynamics.The numerical methods studied in this work include the finite difference methods, exponential wave integrator methods as well as the time-splitting methods and particular attentions are paid on the error bounds of different numerical methods up to the time <span class=\"etd-inline-math\">t= T<sub>0</sub>/\\varepsilon<sup>&beta;</sup></span> with <span class=\"etd-inline-math\">0 \\leq &beta; \\leq 2</span> and <span class=\"etd-inline-math\">T<sub>0</sub></span> fixed. As a by-product, the results are extended to solve an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and <span class=\"etd-inline-math\">O(\\varepsilon<sup>&beta;</sup>)</span> in time. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.","abstract_has_math":true,"creators":["FENG YUE"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-06-10","date_published":"2020-06-10","updated_at":"2026-07-24T03:32:04Z","subjects":["nonlinear Klein-Gordon equation, long-time dynamics, finite difference method, exponential wave integrator, time-splitting method, error estimates"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["FENG YUE"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2020-06-10"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://scholarbank.nus.edu.sg/handle/10635/174472"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["nonlinear Klein-Gordon equation, long-time dynamics, finite difference method, exponential wave integrator, time-splitting method, error estimates"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://scholarbank.nus.edu.sg/bitstreams/eadaca5e-05e9-4f57-93ef-12393e378d6f/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This work is devoted to the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\\varepsilon^2$ with $\\varepsilon \\in (0, 1]$ a dimensionless parameter.The analytical results indicate that the life-span of a smooth solution to this NKGE is at least up to the time at $O(\\varepsilon^{-2})$. Different numerical methods are adapted to discretize the problem and rigorous error bounds are established for the long-time dynamics.The numerical methods studied in this work include the finite difference methods, exponential wave integrator methods as well as the time-splitting methods and particular attentions are paid on the error bounds of different numerical methods up to the time $t= T_0/\\varepsilon^{\\beta}$ with $0 \\leq \\beta \\leq 2$ and $T_0$ fixed. As a by-product, the results are extended to solve an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\\varepsilon^{\\beta})$ in time. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["4275745abb10f29fd0a14f2ec2110bea","f27e7b7e189c078f9accf040dd142d98"]},{"key":"dc:title","label":"Title","values":["ERROR ESTIMATES OF NUMERICAL METHODS FOR THE LONG-TIME DYNAMICS OF THE NONLINEAR KLEIN-GORDON EQUATION"]}]}],"canonical_facts":{"dc:creator":["FENG YUE"],"dc:date.issued":["2020-06-10"],"dc:description.abstract":["This work is devoted to the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\\varepsilon^2$ with $\\varepsilon \\in (0, 1]$ a dimensionless parameter.The analytical results indicate that the life-span of a smooth solution to this NKGE is at least up to the time at $O(\\varepsilon^{-2})$. Different numerical methods are adapted to discretize the problem and rigorous error bounds are established for the long-time dynamics.The numerical methods studied in this work include the finite difference methods, exponential wave integrator methods as well as the time-splitting methods and particular attentions are paid on the error bounds of different numerical methods up to the time $t= T_0/\\varepsilon^{\\beta}$ with $0 \\leq \\beta \\leq 2$ and $T_0$ fixed. As a by-product, the results are extended to solve an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\\varepsilon^{\\beta})$ in time. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp."],"dc:format.checksum.md5":["4275745abb10f29fd0a14f2ec2110bea","f27e7b7e189c078f9accf040dd142d98"],"dc:identifier.uri":["https://scholarbank.nus.edu.sg/bitstreams/eadaca5e-05e9-4f57-93ef-12393e378d6f/download"],"dc:relation.isreferencedby":["https://scholarbank.nus.edu.sg/handle/10635/174472"],"dc:subject":["nonlinear Klein-Gordon equation, long-time dynamics, finite difference method, exponential wave integrator, time-splitting method, error estimates"],"dc:title":["ERROR ESTIMATES OF NUMERICAL METHODS FOR THE LONG-TIME DYNAMICS OF THE NONLINEAR KLEIN-GORDON EQUATION"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T03:32:04Z"}