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<p>We propose a novel high-order compact (HOC) finite difference scheme for solving Helmholtz and diffusion–advection equations under Robin boundary conditions. Derived through successive variable elimination from difference equation systems, the method achieves fourth- and sixth-order convergence for the Helmholtz equation and fourth-order convergence for the diffusion–advection equation. Theoretical analysis establishes solvability and proves error estimates for all schemes. Completed Richardson extrapolation applied to the diffusion–advection solver yields fifth-order accuracy (wavenumber <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K \ne 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and sixth-order accuracy (wavenumber <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(K = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). Numerical experiments confirm theoretical convergence rates for smooth and highly oscillatory solutions.</p>

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High order compact difference scheme for elliptic equations with Robin boundary conditions

  • 期刊论文

  • 发表日期：

    2025年12月18日

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  • Wenjing Liu,
  • Ping Yin

摘要

We propose a novel high-order compact (HOC) finite difference scheme for solving Helmholtz and diffusion–advection equations under Robin boundary conditions. Derived through successive variable elimination from difference equation systems, the method achieves fourth- and sixth-order convergence for the Helmholtz equation and fourth-order convergence for the diffusion–advection equation. Theoretical analysis establishes solvability and proves error estimates for all schemes. Completed Richardson extrapolation applied to the diffusion–advection solver yields fifth-order accuracy (wavenumber \(K \ne 0\) K ≠ 0 ) and sixth-order accuracy (wavenumber \(K = 0\) K = 0 ). Numerical experiments confirm theoretical convergence rates for smooth and highly oscillatory solutions.