A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (O...

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Autores principales:	Jiong Weng, Xiaojing Liu, Youhe Zhou, Jizeng Wang
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	nonlinear wave equations Klein–Gordon equation sine–Gordon equation wavelet integral collocation method space-time fully decoupled formulation Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/039d97ccf29d433f80ac2da9486fbd7d
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spelling	oai:doaj.org-article:039d97ccf29d433f80ac2da9486fbd7d2021-11-25T18:17:33ZA Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations10.3390/math92229572227-7390https://doaj.org/article/039d97ccf29d433f80ac2da9486fbd7d2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2957https://doaj.org/toc/2227-7390A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.Jiong WengXiaojing LiuYouhe ZhouJizeng WangMDPI AGarticlenonlinear wave equationsKlein–Gordon equationsine–Gordon equationwavelet integral collocation methodspace-time fully decoupled formulationMathematicsQA1-939ENMathematics, Vol 9, Iss 2957, p 2957 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	nonlinear wave equations Klein–Gordon equation sine–Gordon equation wavelet integral collocation method space-time fully decoupled formulation Mathematics QA1-939
spellingShingle	nonlinear wave equations Klein–Gordon equation sine–Gordon equation wavelet integral collocation method space-time fully decoupled formulation Mathematics QA1-939 Jiong Weng Xiaojing Liu Youhe Zhou Jizeng Wang A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
description	A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.
format	article
author	Jiong Weng Xiaojing Liu Youhe Zhou Jizeng Wang
author_facet	Jiong Weng Xiaojing Liu Youhe Zhou Jizeng Wang
author_sort	Jiong Weng
title	A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
title_short	A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
title_full	A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
title_fullStr	A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
title_full_unstemmed	A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations
title_sort	space-time fully decoupled wavelet integral collocation method with high-order accuracy for a class of nonlinear wave equations
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/039d97ccf29d433f80ac2da9486fbd7d
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A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

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