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Article

Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
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Fractal Fract. 2025, 9(8), 520; https://doi.org/10.3390/fractalfract9080520
Submission received: 3 July 2025 / Revised: 4 August 2025 / Accepted: 6 August 2025 / Published: 8 August 2025

Abstract

Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. In order to establish the stability and convergence analysis, we introduce a norm||·||H˜1,which is rigorously proved equivalent to the standard H1-norm. Considering that the coefficients of high-order numerical schemes are not entirely positive, we introduce an appropriate parameter to transform the numerical scheme into an equivalent form with positive coefficients. Based on the equivalent form, we prove that the temporal and spatial convergence orders are (3γ) and 4 by applying the convergence of geometric progression. The proposed scheme ensures that the theoretical convergence accuracy at each time step is of order (3γ) without requiring any additional processing techniques. Ultimately, the convergence of the proposed high-order accurate scheme is verified through numerical experiments involving (non-)linear high-dimensional TFDEs.
Keywords: stability and convergence analysis; high-dimensional model; spatial compact finite difference scheme; L2-type scheme; time-fractional diffusion equations stability and convergence analysis; high-dimensional model; spatial compact finite difference scheme; L2-type scheme; time-fractional diffusion equations

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MDPI and ACS Style

Cao, J.-Y.; Fang, J.-Q.; Wang, Z.-Q.; Wang, Z.-Q. Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time. Fractal Fract. 2025, 9, 520. https://doi.org/10.3390/fractalfract9080520

AMA Style

Cao J-Y, Fang J-Q, Wang Z-Q, Wang Z-Q. Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time. Fractal and Fractional. 2025; 9(8):520. https://doi.org/10.3390/fractalfract9080520

Chicago/Turabian Style

Cao, Jun-Ying, Jian-Qiang Fang, Zhong-Qing Wang, and Zi-Qiang Wang. 2025. "Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time" Fractal and Fractional 9, no. 8: 520. https://doi.org/10.3390/fractalfract9080520

APA Style

Cao, J.-Y., Fang, J.-Q., Wang, Z.-Q., & Wang, Z.-Q. (2025). Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time. Fractal and Fractional, 9(8), 520. https://doi.org/10.3390/fractalfract9080520

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