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<p>In this paper, we analyze higher-order weighted and shifted Grünwald-Letnikov (WSGL) schemes for tempered subdiffusion problems with nonsingular and singular source terms. Building on the correction technique, we employ the resolvent estimates of the numerical schemes to demonstrate that conventional high-order WSGL schemes achieve only first-order convergence, regardless of the smoothness of data. Drawing on the insights from Jin et al. (SIAM J. Sci. Comput., 39(6) (2017), A3129-A3152) regarding correction terms, we introduce tailored corrections in the initial steps and integral-differential approach for singular source terms in time, then propose modified higher-order WSGL schemes for the tempered subdiffusion equation. Rigorous analysis shows that our corrected schemes preserve high-order accuracy, attaining optimal convergence orders of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(\tau ^{k})~(k=2,3,4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for both smooth and nonsmooth data. Numerical experiments are provided to validate our theoretical findings.</p>

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Analysis of higher order WSGL schemes for tempered subdiffusion equation with nonsmooth data

  • 期刊论文

  • 发表日期：

    2026年7月23日

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  • Can Li,
  • Xin Wang,
  • Shimin Guo,
  • Wenyi Tian

摘要

In this paper, we analyze higher-order weighted and shifted Grünwald-Letnikov (WSGL) schemes for tempered subdiffusion problems with nonsingular and singular source terms. Building on the correction technique, we employ the resolvent estimates of the numerical schemes to demonstrate that conventional high-order WSGL schemes achieve only first-order convergence, regardless of the smoothness of data. Drawing on the insights from Jin et al. (SIAM J. Sci. Comput., 39(6) (2017), A3129-A3152) regarding correction terms, we introduce tailored corrections in the initial steps and integral-differential approach for singular source terms in time, then propose modified higher-order WSGL schemes for the tempered subdiffusion equation. Rigorous analysis shows that our corrected schemes preserve high-order accuracy, attaining optimal convergence orders of \(O(\tau ^{k})~(k=2,3,4)\) O ( τ k ) ( k = 2 , 3 , 4 ) for both smooth and nonsmooth data. Numerical experiments are provided to validate our theoretical findings.