This research devises a rapid and effective numerical approach tailored to a three-dimensional integro-partial differential equation that encompasses three weakly singular kernels. The second-order convolution quadrature rule is employed to approximate the Riemann-Liouvile integral term in time, and the compact difference scheme is utilized in space. Moreover, the Crank-Nicolson alternating direction implicit method is adopted. The discrete energy method is then used to rigorously prove the solvability, convergence, and stability of the constructed scheme. The convergence order is proved with order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2386_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}(\tau ^{2}+h_{x}^{4}+h_{y}^{4}+h_{z}^{4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>h</mi> <mrow> <mi>x</mi> </mrow> <mn>4</mn> </msubsup> <mo>+</mo> <msubsup> <mi>h</mi> <mrow> <mi>y</mi> </mrow> <mn>4</mn> </msubsup> <mo>+</mo> <msubsup> <mi>h</mi> <mrow> <mi>z</mi> </mrow> <mn>4</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2386_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> represents the time step, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2386_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{x}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2386_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{y}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>y</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2386_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{z}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> are the step sizes in the x, y, z directions, respectively. The ADI algorithm is capable of converting the three-dimensional problem under discussion in this paper into the continuous solution of three one-dimensional problems, thereby circumventing the necessity of solving linear equations with large sparse matrices serving as coefficient matrices. This approach remarkably curtails the computational time. Finally, some numerical examples are provided to verify the accuracy of the theoretical analysis.

The ADI compact difference scheme for three-dimensional integro-partial differential equation with three weakly singular kernels

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发表日期：

2025年1月25日

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Tianyuan Liu,
Haixiang Zhang,
Xuehua Yang

摘要

This research devises a rapid and effective numerical approach tailored to a three-dimensional integro-partial differential equation that encompasses three weakly singular kernels. The second-order convolution quadrature rule is employed to approximate the Riemann-Liouvile integral term in time, and the compact difference scheme is utilized in space. Moreover, the Crank-Nicolson alternating direction implicit method is adopted. The discrete energy method is then used to rigorously prove the solvability, convergence, and stability of the constructed scheme. The convergence order is proved with order ${\mathcal {O}}(\tau ^{2}+h_{x}^{4}+h_{y}^{4}+h_{z}^{4})$ $O (τ^{2} + h_{x}^{4} + h_{y}^{4} + h_{z}^{4})$ , where $\tau $ $τ$ represents the time step, and $h_{x}$ $h_{x}$ , $h_{y}$ $h_{y}$ , $h_{z}$ $h_{z}$ are the step sizes in the x, y, z directions, respectively. The ADI algorithm is capable of converting the three-dimensional problem under discussion in this paper into the continuous solution of three one-dimensional problems, thereby circumventing the necessity of solving linear equations with large sparse matrices serving as coefficient matrices. This approach remarkably curtails the computational time. Finally, some numerical examples are provided to verify the accuracy of the theoretical analysis.