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<p>We present a compact solver for the generalised Kuramoto–Sivashinsky (GKS) equation that combines a global Gaussian kernel smoothing in space with a Crank–Nicolson (CN) time integrator and a conservative Newton treatment of the quadratic flux. Non-periodic plateau boundary data are imposed by clamping a few boundary nodes. We monitor three space integrals (“mass” <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P_1\)</EquationSource> </InlineEquation>, an energy-like <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P_2\)</EquationSource> </InlineEquation>, and a higher-order diagnostic <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P_3\)</EquationSource> </InlineEquation>) and a frozen-operator energy indicator. On two travelling-wave benchmarks the method delivers pointwise errors <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\mathcal {O}}(10^{-4})\)</EquationSource> </InlineEquation> and small weighted relative errors on a modest grid (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(m=601\)</EquationSource> </InlineEquation>), while diagnostics remain flat (dispersive case) or decay gently (dissipative case).</p>

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Numerical approximation of the generalised Kuramoto–Sivashinsky equation by a kernel smoothing technique

  • 期刊论文

  • 发表日期：

    2025年9月4日

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  • Zain Ul Abidin,
  • Shafiq Ur Rehman,
  • Fayyaz Ahmad,
  • Zainab Saeed

摘要

We present a compact solver for the generalised Kuramoto–Sivashinsky (GKS) equation that combines a global Gaussian kernel smoothing in space with a Crank–Nicolson (CN) time integrator and a conservative Newton treatment of the quadratic flux. Non-periodic plateau boundary data are imposed by clamping a few boundary nodes. We monitor three space integrals (“mass” \(P_1\) , an energy-like \(P_2\) , and a higher-order diagnostic \(P_3\) ) and a frozen-operator energy indicator. On two travelling-wave benchmarks the method delivers pointwise errors \({\mathcal {O}}(10^{-4})\) and small weighted relative errors on a modest grid ( \(m=601\) ), while diagnostics remain flat (dispersive case) or decay gently (dissipative case).