{"title":"论求解随机强阻尼波方程的全离散方案的强收敛性","authors":"Chengqiang Xu, Yibo Wang, Wanrong Cao","doi":"10.1002/num.23094","DOIUrl":null,"url":null,"abstract":"This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio-temporal discretization of the approximate equation by a finite element method in space and an exponential trapezoidal scheme in time. We prove that the combination can derive higher strong convergence order in time than the use of the piecewise approximation of the noise and the exponential Euler scheme or the implicit Euler scheme in time. Particularly, the temporal strong convergence order of the fully discrete scheme reaches <mjx-container aria-label=\"5 divided by 4 minus epsilon\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic- data-semantic-role=\"subtraction\" data-semantic-speech=\"5 divided by 4 minus epsilon\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"5\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"6\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/246ca0b6-0dc3-49d5-80f8-c979b4835343/num23094-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic-role=\"subtraction\" data-semantic-speech=\"5 divided by 4 minus epsilon\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">5</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"5\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">4</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,−\" data-semantic-parent=\"6\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">ε</mi></mrow>$$ 5/4-\\varepsilon $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> for the one-dimensional space-time white noise, which overcomes the order barrier one. Moreover, we allow the covariance operator of the noise to be noncommutative with the Dirichlet Laplacian, which weakens the common assumptions on the noise in the literature. Finally, some numerical experiments in different spatial dimensions are presented to support our theoretical findings. By means of the piecewise spectral approximation of the noise, a piecewise version of the fully discrete scheme is constructed to fulfill a long-time simulation.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"199 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations\",\"authors\":\"Chengqiang Xu, Yibo Wang, Wanrong Cao\",\"doi\":\"10.1002/num.23094\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio-temporal discretization of the approximate equation by a finite element method in space and an exponential trapezoidal scheme in time. We prove that the combination can derive higher strong convergence order in time than the use of the piecewise approximation of the noise and the exponential Euler scheme or the implicit Euler scheme in time. Particularly, the temporal strong convergence order of the fully discrete scheme reaches <mjx-container aria-label=\\\"5 divided by 4 minus epsilon\\\" ctxtmenu_counter=\\\"0\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"5,4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-role=\\\"subtraction\\\" data-semantic-speech=\\\"5 divided by 4 minus epsilon\\\" data-semantic-type=\\\"infixop\\\"><mjx-mrow data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" space=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,−\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/246ca0b6-0dc3-49d5-80f8-c979b4835343/num23094-math-0001.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"5,4\\\" data-semantic-content=\\\"3\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-speech=\\\"5 divided by 4 minus epsilon\\\" data-semantic-type=\\\"infixop\\\"><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,2\\\" data-semantic-content=\\\"1\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">5</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" stretchy=\\\"false\\\">/</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">4</mn></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,−\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" form=\\\"prefix\\\">−</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">ε</mi></mrow>$$ 5/4-\\\\varepsilon $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> for the one-dimensional space-time white noise, which overcomes the order barrier one. Moreover, we allow the covariance operator of the noise to be noncommutative with the Dirichlet Laplacian, which weakens the common assumptions on the noise in the literature. Finally, some numerical experiments in different spatial dimensions are presented to support our theoretical findings. By means of the piecewise spectral approximation of the noise, a piecewise version of the fully discrete scheme is constructed to fulfill a long-time simulation.\",\"PeriodicalId\":19443,\"journal\":{\"name\":\"Numerical Methods for Partial Differential Equations\",\"volume\":\"199 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-02-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerical Methods for Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/num.23094\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Methods for Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/num.23094","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations
This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio-temporal discretization of the approximate equation by a finite element method in space and an exponential trapezoidal scheme in time. We prove that the combination can derive higher strong convergence order in time than the use of the piecewise approximation of the noise and the exponential Euler scheme or the implicit Euler scheme in time. Particularly, the temporal strong convergence order of the fully discrete scheme reaches for the one-dimensional space-time white noise, which overcomes the order barrier one. Moreover, we allow the covariance operator of the noise to be noncommutative with the Dirichlet Laplacian, which weakens the common assumptions on the noise in the literature. Finally, some numerical experiments in different spatial dimensions are presented to support our theoretical findings. By means of the piecewise spectral approximation of the noise, a piecewise version of the fully discrete scheme is constructed to fulfill a long-time simulation.
期刊介绍:
An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.