In this paper, the geometrically increased time step sizes (GITSS) are proposed to discretize the Caputo fractional derivative in the subdiffusion equation. The coefficient matrix of the resulting all-at-once linear system is found to be a block lower triangular Toeplitz-like matrix. We combine the divide-and-conquer strategy with the block forward substitution method to solve this linear system efficiently. The computational cost of the proposed method is of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {O}(MN\log ^2N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mi>N</mi> <msup> <mo>log</mo> <mn>2</mn> </msup> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where M and N are the degrees of freedom in space and time, respectively. The numerical experiments report that the CPU times of the proposed method are much less than those of the popular exponential-sum-approximation (ESA) method in the references. Moreover, we prove that the L1 scheme with the GITSS possesses a convergence order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathscr {O}(\frac{\ln ^{\max \{\sigma ,2-\alpha \}}N}{N^{2-\alpha }})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mfrac> <mrow> <msup> <mo>ln</mo> <mrow> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mi>σ</mi> <mo>,</mo> <mn>2</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> </msup> <mi>N</mi> </mrow> <msup> <mi>N</mi> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in temporal direction if the time mesh parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma \ge \max \{0,\frac{2-\alpha }{\sigma }-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>≥</mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> <mi>σ</mi> </mfrac> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> reflects the regularity of the solution. Although under the GITSS the convergence order is slightly less than that of the graded mesh, the Toeplitz-like structure brought by the GITSS has more potential to develop the fast algorithm.

All-at-once Implementation For Subdiffusion Equation With Time Nonuniform Mesh

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发表日期：

2026年7月11日

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Tao Sun,
Tian-Yi Li,
Hai-Wei Sun

摘要

In this paper, the geometrically increased time step sizes (GITSS) are proposed to discretize the Caputo fractional derivative in the subdiffusion equation. The coefficient matrix of the resulting all-at-once linear system is found to be a block lower triangular Toeplitz-like matrix. We combine the divide-and-conquer strategy with the block forward substitution method to solve this linear system efficiently. The computational cost of the proposed method is of $\mathscr {O}(MN\log ^2N)$ $O (M N {log}^{2} N)$ , where M and N are the degrees of freedom in space and time, respectively. The numerical experiments report that the CPU times of the proposed method are much less than those of the popular exponential-sum-approximation (ESA) method in the references. Moreover, we prove that the L1 scheme with the GITSS possesses a convergence order of $\mathscr {O}(\frac{\ln ^{\max \{\sigma ,2-\alpha \}}N}{N^{2-\alpha }})$ $O (\frac{{ln}^{max {σ, 2 - α}} N}{N^{2 - α}})$ in temporal direction if the time mesh parameter $\gamma \ge \max \{0,\frac{2-\alpha }{\sigma }-1\}$ $γ \geq max {0, \frac{2 - α}{σ} - 1}$ , where $\sigma $ $σ$ reflects the regularity of the solution. Although under the GITSS the convergence order is slightly less than that of the graded mesh, the Toeplitz-like structure brought by the GITSS has more potential to develop the fast algorithm.