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<p>In this paper, we study a numerical method for the Caputo time fractional wave equation with nonsmooth data. We first introduce a class of third-order approximations, known as weighted and shifted Grünwald-Letnikov approximations, to approximate the Caputo fractional derivative. Based on this, we develop a new time stepping method for solving the time fractional wave equation. After applying corrections to several initial steps, the proposed time stepping method achieves a convergence order of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(k^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for nonsmooth data, where <i>k</i> denotes the time step size. We also analyze the stability regions of the proposed time stepping method, which show that the scheme is unconditionally stable for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \alpha \in (1, 1.94) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1.94</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and conditionally stable for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \alpha \in [1.94, 2) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1.94</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Numerical experiments are presented to validate the theoretical findings.</p>

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Third-Order Time Stepping Methods for Superdiffusion Using Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • Yonghui He,
  • Jinghua Chen,
  • Yubin Yan,
  • Xuejuan Chen,
  • Xinran Liu,
  • Peng Ding

摘要

In this paper, we study a numerical method for the Caputo time fractional wave equation with nonsmooth data. We first introduce a class of third-order approximations, known as weighted and shifted Grünwald-Letnikov approximations, to approximate the Caputo fractional derivative. Based on this, we develop a new time stepping method for solving the time fractional wave equation. After applying corrections to several initial steps, the proposed time stepping method achieves a convergence order of \(O(k^3)\) O ( k 3 ) for nonsmooth data, where k denotes the time step size. We also analyze the stability regions of the proposed time stepping method, which show that the scheme is unconditionally stable for \( \alpha \in (1, 1.94) \) α ∈ ( 1 , 1.94 ) , and conditionally stable for \( \alpha \in [1.94, 2) \) α ∈ [ 1.94 , 2 ) . Numerical experiments are presented to validate the theoretical findings.