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<p>In this paper, we consider the Serre–Green–Naghdi equations over uneven bottoms and with surface tension effects. We focus on two asymptotic models derived from them: the Boussinesq and Camassa–Holm models. In particular, we study their simplified forms, denoted by KdV<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> and CH<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, which correspond to the Korteweg–de Vries and Camassa–Holm equations, respectively. Our study involves a numerical investigation of finite difference schemes that conserve discrete energy in fully discrete formulations for both models. The objective is to compute accurate approximate solutions using efficient Python implementations that evolve the solutions recursively in time <i>t</i> and space <i>x</i>, and to visualize them in both two and three dimensions.</p>

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A Numerical Study of KDV and CH Equations with Surface Tension

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Mohammad Haidar,
  • Toufic El Arwadi,
  • Samer Israwi,
  • Charbel Geryes Aoun

摘要

In this paper, we consider the Serre–Green–Naghdi equations over uneven bottoms and with surface tension effects. We focus on two asymptotic models derived from them: the Boussinesq and Camassa–Holm models. In particular, we study their simplified forms, denoted by KdV \(\sigma \) σ and CH \(\sigma \) σ , which correspond to the Korteweg–de Vries and Camassa–Holm equations, respectively. Our study involves a numerical investigation of finite difference schemes that conserve discrete energy in fully discrete formulations for both models. The objective is to compute accurate approximate solutions using efficient Python implementations that evolve the solutions recursively in time t and space x, and to visualize them in both two and three dimensions.