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<p>The weighted and shifted backward differentiation formula (WSBDF) with a weighted parameter is an improved BDF-type method introduced by Akrivis et al. [IMA J. Numer. Anal., 45(6):3207-3234, 2025]. By introducing the weighted parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, it not only preserves the stability of the classical BDF method but also enhances the stability region and flexibility of the method. This paper presents a class of WSBDF schemes up to the fifth order for semilinear parabolic equations. Unlike the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] and explicit uniform multiplier technique [SIAM J. Numer. Anal., 62(4):1609-1637, 2024] based on Dahlquist’s G-stability theory [BIT, 18:384-401, 1978], we apply the discrete orthogonal convolution kernels technique to propose a straightforward discrete energy method and establish <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm stability as well as error estimates for the high-order WSBDF schemes. Finally, numerical experiments are conducted to support our theoretical results.</p>

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A straightforward discrete energy technique for weighted and shifted backward differentiation formulas up to fifth order

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Yuanyuan Kang,
  • Dongqian Li,
  • Yin Yang

摘要

The weighted and shifted backward differentiation formula (WSBDF) with a weighted parameter is an improved BDF-type method introduced by Akrivis et al. [IMA J. Numer. Anal., 45(6):3207-3234, 2025]. By introducing the weighted parameter \(\beta \) β , it not only preserves the stability of the classical BDF method but also enhances the stability region and flexibility of the method. This paper presents a class of WSBDF schemes up to the fifth order for semilinear parabolic equations. Unlike the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] and explicit uniform multiplier technique [SIAM J. Numer. Anal., 62(4):1609-1637, 2024] based on Dahlquist’s G-stability theory [BIT, 18:384-401, 1978], we apply the discrete orthogonal convolution kernels technique to propose a straightforward discrete energy method and establish \(L^2\) L 2 norm stability as well as error estimates for the high-order WSBDF schemes. Finally, numerical experiments are conducted to support our theoretical results.