Fourth-Order Scheme for Riesz Tempered Fractional Diffusion

Researchers Xiaotong Li, Taiqiang Jiang, and Zeshan Qiu have proposed a high-order numerical scheme designed for the spatial Riesz tempered fractional diffusion equation. This mathematical framework is used to characterize transient anomalous diffusion phenomena that involve exponentially tempered power-law jumps. The study was published in the journal Boundary Value Problems.

The core innovation of the work involves the introduction of a tempered compact differential operator. By utilizing this operator, the authors constructed a fourth-order compact difference approximation for the Riesz tempered fractional derivative. This approach addresses diffusion processes characterized by exponentially tempered power-law jumps.

For the temporal discretization, the scheme employs the (2,2) Padé approximation. This choice of method allows the fully discrete scheme to achieve fourth-order accuracy in both space and time. Such high-order accuracy is achieved while maintaining stability and convergence properties.

The stability and convergence of the proposed scheme were established using the energy method.

Numerical experiments were conducted to confirm the effectiveness and advancement of the proposed scheme. These experiments confirm the effectiveness and advancement of the proposed scheme. The paper is available as an open access article.

Subject of Research: Numerical methods for fractional diffusion equations

Article Title: A fourth-order numerical scheme for the spatial Riesz tempered fractional diffusion equation

Article References: Li, X., Jiang, T., & Qiu, Z. (2026). A fourth-order numerical scheme for the spatial Riesz tempered fractional diffusion equation. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02359-w

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DOI: 10.1186/s13661-026-02359-w

Keywords: fractional diffusion, numerical scheme, Riesz derivative, tempered jumps, Padé approximation, anomalous diffusion, fourth-order accuracy, energy method, fourth-order, numerical, scheme, spatial