<p>In this paper, the backward Euler method and the shifted Grünwald-Letnikov formulas are utilized to discretize the space-fractional diffusion equations. The discretized result is a system of linear equations with a coefficient matrix being the sum of a diagonal matrix and a non-Hermitian Toeplitz matrix. By utilizing the Hermitian and skew-Hermitian splitting of the Toeplitz matrix, we develop a two-parameter DT<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_491_Article_IEq1.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$_{\textrm{HS}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mtext>HS</mtext> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>S iteration method to solve the linear systems. The convergence is also discussed. A DT<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_491_Article_IEq1.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$_{\textrm{HS}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mtext>HS</mtext> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>S-<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_491_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau (\alpha ,\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> preconditioner is proposed and the preconditioned GMRES method combined with the proposed preconditioner is applied to solve the linear systems. The spectral analysis of the DT<InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_491_Article_IEq1.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$_{\textrm{HS}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mtext>HS</mtext> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>S-<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_491_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau (\alpha ,\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.</p>

A DT$_{\textrm{HS}}$S-$\tau $ Preconditioner for the Discretized Linear Systems of Space-Fractional Diffusion Equations

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发表日期：

2025年6月17日

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Shi-Ping Tang,
Yu-Mei Huang

摘要

In this paper, the backward Euler method and the shifted Grünwald-Letnikov formulas are utilized to discretize the space-fractional diffusion equations. The discretized result is a system of linear equations with a coefficient matrix being the sum of a diagonal matrix and a non-Hermitian Toeplitz matrix. By utilizing the Hermitian and skew-Hermitian splitting of the Toeplitz matrix, we develop a two-parameter DT $_{\textrm{HS}}$ $_{HS}$ S iteration method to solve the linear systems. The convergence is also discussed. A DT $_{\textrm{HS}}$ $_{HS}$ S- $\tau (\alpha ,\gamma )$ $τ (α, γ)$ preconditioner is proposed and the preconditioned GMRES method combined with the proposed preconditioner is applied to solve the linear systems. The spectral analysis of the DT $_{\textrm{HS}}$ $_{HS}$ S- $\tau (\alpha ,\gamma )$ $τ (α, γ)$ preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.

A DT\(_{\textrm{HS}}\)S-\(\tau \) Preconditioner for the Discretized Linear Systems of Space-Fractional Diffusion Equations

摘要