Numerical solutions of the EW and MEW equations using a fourth-order improvised B-spline collocation method

A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretizatio...

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Vydáno v:	Numerical algorithms Ročník 98; číslo 4; s. 1799 - 1825
Hlavní autoři:	Fan, Guangyu, Wu, Beibei
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2025 Springer Nature B.V
Témata:	Algebra Algorithms Approximation B spline functions Boundary conditions Collocation methods Computer Science Crank-Nicholson method Discretization Exact solutions Fourier series Mathematical analysis Methods Norms Numeric Computing Numerical Analysis Original Paper Partial differential equations Solitary waves Stability analysis Theory of Computation Wave propagation Stability EW and MEW equations Gauss elimination method Cubic B-spline method Crank-Nicolson scheme
ISSN:	1017-1398, 1572-9265
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Abstract	A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretization of the temporal domain. The nonlinear terms are processed using quasi-linearization techniques and the stability analysis of this method is performed using Fourier series analysis. The validity and accuracy of this method are verified through several numerical experiments using a single solitary wave, two solitary waves, Maxwellian initial condition, and an undular bore. Since there is an exact solution for the single wave, the error norms L 2 and L ∞ are first calculated and compared with some previous studies published in journal articles. In addition, the three conserved quantities Q , M , and E of the problems raised during the simulation are also calculated and recorded in the table. Lastly, the comparisons of these error norms and conserved quantities show that the numerical results obtained with the proposed method are more accurate and agree well with the values of the conserved quantities obtained in some literatures using the same parameters. The main advantage of ICSCM is its ability to effectively capture solitary wave propagation and describe solitary wave collisions. It can perform solution calculations at any point in the domain, easily use larger time steps to calculate solutions at higher time levels, and produce more accurate calculation results.
AbstractList	A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretization of the temporal domain. The nonlinear terms are processed using quasi-linearization techniques and the stability analysis of this method is performed using Fourier series analysis. The validity and accuracy of this method are verified through several numerical experiments using a single solitary wave, two solitary waves, Maxwellian initial condition, and an undular bore. Since there is an exact solution for the single wave, the error norms L 2 and L ∞ are first calculated and compared with some previous studies published in journal articles. In addition, the three conserved quantities Q , M , and E of the problems raised during the simulation are also calculated and recorded in the table. Lastly, the comparisons of these error norms and conserved quantities show that the numerical results obtained with the proposed method are more accurate and agree well with the values of the conserved quantities obtained in some literatures using the same parameters. The main advantage of ICSCM is its ability to effectively capture solitary wave propagation and describe solitary wave collisions. It can perform solution calculations at any point in the domain, easily use larger time steps to calculate solutions at higher time levels, and produce more accurate calculation results. A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretization of the temporal domain. The nonlinear terms are processed using quasi-linearization techniques and the stability analysis of this method is performed using Fourier series analysis. The validity and accuracy of this method are verified through several numerical experiments using a single solitary wave, two solitary waves, Maxwellian initial condition, and an undular bore. Since there is an exact solution for the single wave, the error norms L2 and L∞ are first calculated and compared with some previous studies published in journal articles. In addition, the three conserved quantities Q, M, and E of the problems raised during the simulation are also calculated and recorded in the table. Lastly, the comparisons of these error norms and conserved quantities show that the numerical results obtained with the proposed method are more accurate and agree well with the values of the conserved quantities obtained in some literatures using the same parameters. The main advantage of ICSCM is its ability to effectively capture solitary wave propagation and describe solitary wave collisions. It can perform solution calculations at any point in the domain, easily use larger time steps to calculate solutions at higher time levels, and produce more accurate calculation results.
Author	Wu, Beibei Fan, Guangyu
Author_xml	– sequence: 1 givenname: Guangyu surname: Fan fullname: Fan, Guangyu organization: School of Mathematics and Physics, Shanghai University of Electric Power – sequence: 2 givenname: Beibei surname: Wu fullname: Wu, Beibei email: beibei.wu@shiep.edu.cn organization: School of Mathematics and Physics, Shanghai University of Electric Power
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Cites_doi	10.1016/j.chaos.2007.06.104 10.1016/S0045-7825(99)00312-6 10.1137/0711049 10.19139/soic.v4i1.167 10.1016/j.cnsns.2004.07.001 10.1002/num.22470 10.1016/j.amc.2004.08.013 10.1080/00207160600740958 10.1016/0009-2509(66)85007-8 10.1017/S0022112066001678 10.1016/0375-9601(76)90714-3 10.1155/2012/587208 10.4236/am.2016.710102 10.1098/rsta.1972.0032 10.1006/jcph.1994.1113 10.1016/j.amc.2011.08.013 10.1002/(SICI)1099-0887(199707)13 10.1142/S0129183119500876 10.1016/S0010-4655(99)00471-3 10.36045/bbms/1337864268 10.1016/0167-2789(84)90014-9 10.4236/jamp.2016.46110 10.1016/j.amc.2005.07.034 10.4236/am.2011.26098 10.1016/j.amc.2003.08.105 10.1016/0021-9991(92)90054-3 10.1134/S0965542515030070
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Snippet	A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width...
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SubjectTerms	Algebra Algorithms Approximation B spline functions Boundary conditions Collocation methods Computer Science Crank-Nicholson method Discretization Exact solutions Fourier series Mathematical analysis Methods Norms Numeric Computing Numerical Analysis Original Paper Partial differential equations Solitary waves Stability analysis Theory of Computation Wave propagation
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Title	Numerical solutions of the EW and MEW equations using a fourth-order improvised B-spline collocation method
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