<p>We present a fitted mesh-based and sum-of-exponentials (SOE) accelerated Alikhanov compact alternating direction implicit (ADI) scheme for solving multi-term and distributed-order two-dimensional time-fractional reaction-diffusion equations (TFRDE) exhibiting weak initial singularities. A central contribution of this work is the construction of a high-order Alikhanov-type approximation for the Caputo fractional derivative, based on a variable super-convergent point that adapts to general non-uniform temporal meshes. To capture the initial-time singularity, we employ a fitted time discretization and establish local truncation error bounds for low-regularity solutions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C[0,T]\cap C^3(0,T]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>∩</mo> <msup> <mi>C</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with a regularity parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> making the framework extendable to distributed-order sub-diffusion models. The resulting scheme achieves the temporal order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$3-\bar{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>-</mo> <mover accent="true"> <mrow> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation> on fitted meshes for multi-term initial-value problem, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \bar{\alpha } $</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> is the maximum fractional order. Leveraging this approximation, we formulate compact ADI schemes for both multi-term and distributed-order TFRDE, and further accelerate them using the SOE technique to reduce the inherited computational and memory complexity. Perturbation terms arising in the ADI formulations are carefully bounded via auxiliary lemmas. Stability and convergence of the presented schemes are rigorously established in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ L_2 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-norm using a discrete fractional Grönwall inequality. The proposed fully-discrete ADI schemes achieve a temporal accuracy of order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ 2\bar{\alpha } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mover accent="true"> <mrow> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>, and fourth-order spatial accuracy. Extensive numerical experiments validate the theory and demonstrate the efficiency of the proposed method.</p>

Fitted Mesh-Based and SOE-Accelerated Alikhanov Compact ADI Schemes for Multi-Term and Distributed-Order 2D Time-Fractional Reaction-Diffusion Equations

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发表日期：

2026年6月23日

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Priyanka,
Sunil Kumar

摘要

We present a fitted mesh-based and sum-of-exponentials (SOE) accelerated Alikhanov compact alternating direction implicit (ADI) scheme for solving multi-term and distributed-order two-dimensional time-fractional reaction-diffusion equations (TFRDE) exhibiting weak initial singularities. A central contribution of this work is the construction of a high-order Alikhanov-type approximation for the Caputo fractional derivative, based on a variable super-convergent point that adapts to general non-uniform temporal meshes. To capture the initial-time singularity, we employ a fitted time discretization and establish local truncation error bounds for low-regularity solutions in $C[0,T]\cap C^3(0,T]$ $C [0, T] \cap C^{3} (0, T]$ , with a regularity parameter $\theta \in (0,1)$ $θ \in (0, 1)$ making the framework extendable to distributed-order sub-diffusion models. The resulting scheme achieves the temporal order $3-\bar{\alpha }$ $3 - \bar{α}$ on fitted meshes for multi-term initial-value problem, where $ \bar{\alpha } $ $\bar{α}$ is the maximum fractional order. Leveraging this approximation, we formulate compact ADI schemes for both multi-term and distributed-order TFRDE, and further accelerate them using the SOE technique to reduce the inherited computational and memory complexity. Perturbation terms arising in the ADI formulations are carefully bounded via auxiliary lemmas. Stability and convergence of the presented schemes are rigorously established in the $ L_2 $ $L_{2}$ -norm using a discrete fractional Grönwall inequality. The proposed fully-discrete ADI schemes achieve a temporal accuracy of order $ 2\bar{\alpha } $ $2 \bar{α}$ , and fourth-order spatial accuracy. Extensive numerical experiments validate the theory and demonstrate the efficiency of the proposed method.