<p>The generalized Rosenau–Kawahara-RLW equation is of great significance in describing the propagation and interaction of waves in physical phenomena. In this work, a high-order compact energy-preserving difference scheme is designed for the generalized Rosenau–Kawahara-RLW equation, which maintains the physical conservation properties of the original problem. In the proposed scheme, we used the Crank–Nicolson method for temporal discretisation and a high-order difference method for spatial discretisation. According to the discrete energy conservation law, the prior estimate of the difference scheme is obtained. By the fixed point theorem and the Gronwall inequality, the existence and uniqueness of the numerical solution are obtained. Based on the prior estimate, the new scheme is proved to be convergent and stable, and the convergence speed of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_945_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$O(\tau ^{2}+h^{4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>h</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is achieved under the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_945_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-norm. The conservation and convergence of the theoretical analysis are verified by numerical experiments, and high numerical efficiency is demonstrated.</p>

A New Compact Energy-Preserving Difference Scheme for the Generalized Rosenau–Kawahara-RLW Equation

期刊论文
发表日期：

2025年1月30日

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Longjie Lv,
Shuguang Li

摘要

The generalized Rosenau–Kawahara-RLW equation is of great significance in describing the propagation and interaction of waves in physical phenomena. In this work, a high-order compact energy-preserving difference scheme is designed for the generalized Rosenau–Kawahara-RLW equation, which maintains the physical conservation properties of the original problem. In the proposed scheme, we used the Crank–Nicolson method for temporal discretisation and a high-order difference method for spatial discretisation. According to the discrete energy conservation law, the prior estimate of the difference scheme is obtained. By the fixed point theorem and the Gronwall inequality, the existence and uniqueness of the numerical solution are obtained. Based on the prior estimate, the new scheme is proved to be convergent and stable, and the convergence speed of $O(\tau ^{2}+h^{4})$ $O (τ^{2} + h^{4})$ is achieved under the $L^{\infty }$ $L^{\infty}$ -norm. The conservation and convergence of the theoretical analysis are verified by numerical experiments, and high numerical efficiency is demonstrated.