It is well-known that designing efficient, high-order, and stable numerical schemes for time non-local differential equations faces three key challenges: non-locality, low solution regularity, and long-term simulation. Achieving this while minimizing storage costs is particularly difficult, especially when attempting to address all three issues simultaneously. In this work, we propose a novel class of numerical schemes designed to simultaneously address the three key challenges associated with time fractional differential equations (TFDEs). In particular, we first derive an equivalent integer-order parametric differential equation (EPDE) for TFDEs, following the idea in [Adv. Nonlinear Anal., 12 (2023), 20220262]. We then establish the stability of EPDEs and provide a rigorous analysis showing that the EPDE exhibits high regularity in the extended <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-direction. Consequently, we employ a Jacobi spectral collocation method (JSCM) for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-direction and, via characteristic decomposition, obtain M independent integer-order ODEs, where M is the number of nodes of JSCM. We then apply the BDF-k (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k=1,\ldots ,5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>) schemes for the time discretization and conduct a rigorous stability analysis. Additionally, we provide an error estimate for the fully discretization scheme, showing a convergence order of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(\Delta t^{k} + M^{-m})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mi>k</mi> </msup> <mo>+</mo> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The computational cost and storage requirements of our proposed algorithm are essentially the same as those for ODEs, namely, O(N) cost and O(1) storage, where N is the total number of time steps. We present several numerical examples to demonstrate the effectiveness of the proposed method and to validate the theoretical results.

A Novel Class of Arbitrary High-order Numerical Schemes for Fractional Differential Equations

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发表日期：

2025年11月5日

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Peng Ding,
Zhiping Mao

摘要

It is well-known that designing efficient, high-order, and stable numerical schemes for time non-local differential equations faces three key challenges: non-locality, low solution regularity, and long-term simulation. Achieving this while minimizing storage costs is particularly difficult, especially when attempting to address all three issues simultaneously. In this work, we propose a novel class of numerical schemes designed to simultaneously address the three key challenges associated with time fractional differential equations (TFDEs). In particular, we first derive an equivalent integer-order parametric differential equation (EPDE) for TFDEs, following the idea in [Adv. Nonlinear Anal., 12 (2023), 20220262]. We then establish the stability of EPDEs and provide a rigorous analysis showing that the EPDE exhibits high regularity in the extended $\theta $ $θ$ -direction. Consequently, we employ a Jacobi spectral collocation method (JSCM) for the $\theta $ $θ$ -direction and, via characteristic decomposition, obtain M independent integer-order ODEs, where M is the number of nodes of JSCM. We then apply the BDF-k ( $k=1,\ldots ,5$ $k = 1, \dots, 5$ ) schemes for the time discretization and conduct a rigorous stability analysis. Additionally, we provide an error estimate for the fully discretization scheme, showing a convergence order of $O(\Delta t^{k} + M^{-m})$ $O (Δ t^{k} + M^{- m})$ . The computational cost and storage requirements of our proposed algorithm are essentially the same as those for ODEs, namely, O(N) cost and O(1) storage, where N is the total number of time steps. We present several numerical examples to demonstrate the effectiveness of the proposed method and to validate the theoretical results.