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<p>In this expository paper, finite difference methods and optimal quadratic and cubic spline collocation schemes for solving both second–order two–point boundary value problems and for the spatial discretization of parabolic initial–boundary value problems are presented in a novel unifying format. In each case Dirichlet, Neumann and periodic boundary conditions are considered. The methods are compact, requiring the solution of tridiagonal linear systems The results of numerical experiments demonstrate their efficacy.</p>

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A unifying framework for compact fourth–order finite difference and spline collocation methods for one–dimensional steady and unsteady convection–diffusion problems

  • 期刊论文

  • 发表日期：

    2026年7月25日

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  • Graeme Fairweather,
  • Andreas Karageorghis

摘要

In this expository paper, finite difference methods and optimal quadratic and cubic spline collocation schemes for solving both second–order two–point boundary value problems and for the spatial discretization of parabolic initial–boundary value problems are presented in a novel unifying format. In each case Dirichlet, Neumann and periodic boundary conditions are considered. The methods are compact, requiring the solution of tridiagonal linear systems The results of numerical experiments demonstrate their efficacy.