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<p>We present a method for computing local compact finite difference (FD) formulations by combining polynomials and radial basis functions (RBFs). We explore two different methods for computing compact FD differentiation weights. The first method involves combining infinitely smooth Multiquadric (MQ) and Gaussian (GA) RBFs with polynomials, while the second method combines polyharmonic spline (PHS) with polynomials. Although we proposed the approach for a general node layouts, for specific stencils, we derive analytical weights for the first and second derivatives and use them to establish the truncation error equations. Our analysis shows that combining MQ or GA with polynomials yields compact FD formulas of the highest possible order on several important specific stencils. Ultimately, the computational outcomes underscore the efficacy and applicability of the presented methods.</p>

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A unified framework for high-order compact finite differences using infinitely- and piecewise-smooth RBFs with polynomials

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Tao Liu,
  • Runqi Xue,
  • Mahdiar Barfeie

摘要

We present a method for computing local compact finite difference (FD) formulations by combining polynomials and radial basis functions (RBFs). We explore two different methods for computing compact FD differentiation weights. The first method involves combining infinitely smooth Multiquadric (MQ) and Gaussian (GA) RBFs with polynomials, while the second method combines polyharmonic spline (PHS) with polynomials. Although we proposed the approach for a general node layouts, for specific stencils, we derive analytical weights for the first and second derivatives and use them to establish the truncation error equations. Our analysis shows that combining MQ or GA with polynomials yields compact FD formulas of the highest possible order on several important specific stencils. Ultimately, the computational outcomes underscore the efficacy and applicability of the presented methods.