<p>This research proposes the numerical solution of the regularized long wave equation using the collocation method with redefined uniform hyperbolic polynomial B-spline as the basis function. The equation is discretized using the Crank-Nicolson method for spatial derivatives and the forward finite difference method for temporal derivatives. Furthermore, the non-linear terms are linearized using the quasilinearization method. The stability of the scheme is determined using the von Neumann method, revealing that the proposed approach is unconditionally stable. The method’s convergence is analyzed through theoretical investigation and numerical experimentation, and it is found to exhibit second order convergence. The method’s versatility is demonstrated by generating single solitary waves, including their interactions in pairs and triads, and the evolution of the Maxwellian initial condition into solitary waves. The three conservation constants are computed and discovered to be closely parallel to the analytical values. Additionally, the convergence of the numerical solution to the exact solution is assessed by calculating the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2573_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({L_2}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2573_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\({L_\infty }\)</EquationSource> </InlineEquation> error norms. The conservation quantities and error norms obtained using the proposed scheme are compared with the outcomes discovered in the literature. The conservation laws reveal that the solitary waves maintain temporal and spatial translational symmetry during propagation, whereas the error norms indicate that the proposed technique provides highly accurate approximate solutions compared to existing methods. To the best of our knowledge, this is the first time the collocation method with redefined uniform hyperbolic polynomial B-spline basis function has been applied to solve the regularized long wave equation. This novel approach provides accurate approximate solutions, making it a promising framework for solving other non-linear partial differential equations.</p>

Numerical simulation of regularized long wave equation using uniform hyperbolic polynomial B-spline based collocation method

期刊论文
发表日期：

2025年6月25日

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Susan Ishwarya A,
Rachna Bhatia

摘要

This research proposes the numerical solution of the regularized long wave equation using the collocation method with redefined uniform hyperbolic polynomial B-spline as the basis function. The equation is discretized using the Crank-Nicolson method for spatial derivatives and the forward finite difference method for temporal derivatives. Furthermore, the non-linear terms are linearized using the quasilinearization method. The stability of the scheme is determined using the von Neumann method, revealing that the proposed approach is unconditionally stable. The method’s convergence is analyzed through theoretical investigation and numerical experimentation, and it is found to exhibit second order convergence. The method’s versatility is demonstrated by generating single solitary waves, including their interactions in pairs and triads, and the evolution of the Maxwellian initial condition into solitary waves. The three conservation constants are computed and discovered to be closely parallel to the analytical values. Additionally, the convergence of the numerical solution to the exact solution is assessed by calculating the \({L_2}\) and \({L_\infty }\) error norms. The conservation quantities and error norms obtained using the proposed scheme are compared with the outcomes discovered in the literature. The conservation laws reveal that the solitary waves maintain temporal and spatial translational symmetry during propagation, whereas the error norms indicate that the proposed technique provides highly accurate approximate solutions compared to existing methods. To the best of our knowledge, this is the first time the collocation method with redefined uniform hyperbolic polynomial B-spline basis function has been applied to solve the regularized long wave equation. This novel approach provides accurate approximate solutions, making it a promising framework for solving other non-linear partial differential equations.