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<p>In this paper, an efficient numerical scheme for solving Burger’s equation has been developed using the uniform hyperbolic polynomial (UHP) B-spline collocation method. This method employs fourth-order UHP B-spline functions as basis functions for approximating the spatial variable, while the Crank-Nicolson scheme is utilized to approximate the time derivative. The stability of the scheme is verified using von Neumann’s criterion, and convergence analysis is also presented. The method has been tested on seven different homogeneous problems and one non-homogeneous problem, with results presented in tables and figures, and compared to existing methods in the literature. The comparison indicates that the proposed method yields more accurate results than other methods. Furthermore, the numerical investigation demonstrates that the method is easy to implement and cost-effective, with a numerically determined rate of convergence close to 2. The proposed method also provides results for small perturbation parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2390_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\((\epsilon \rightarrow 0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where the exact solution exhibits oscillations.</p>

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Efficient numerical solution of Burgers’ equation using collocation method based on re-defined uniform hyperbolic polynomial B-splines

  • 期刊论文

  • 发表日期：

    2025年2月10日

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  • Mansi Palav,
  • Vikas Pradhan

摘要

In this paper, an efficient numerical scheme for solving Burger’s equation has been developed using the uniform hyperbolic polynomial (UHP) B-spline collocation method. This method employs fourth-order UHP B-spline functions as basis functions for approximating the spatial variable, while the Crank-Nicolson scheme is utilized to approximate the time derivative. The stability of the scheme is verified using von Neumann’s criterion, and convergence analysis is also presented. The method has been tested on seven different homogeneous problems and one non-homogeneous problem, with results presented in tables and figures, and compared to existing methods in the literature. The comparison indicates that the proposed method yields more accurate results than other methods. Furthermore, the numerical investigation demonstrates that the method is easy to implement and cost-effective, with a numerically determined rate of convergence close to 2. The proposed method also provides results for small perturbation parameters \((\epsilon \rightarrow 0)\) ( ϵ → 0 ) where the exact solution exhibits oscillations.