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A local RBFs-based DQ approximation for Riesz fractional derivatives and its applications

Abstract

The mathematical models built based on space-fractional derivatives adequately describe the phenomena incorporating spatial heterogeneity, but their numerical simulation on general domains remains a difficult issue. In this study, we develop a local differential quadrature (DQ) formula based on radial basis functions (RBFs) to discretize Riesz fractional derivatives, which is defined by the functional values at the nodes located in the subdomain around the discrete node. The proposed formula overcomes the drawback of ill-conditioning associated with the global DQ method and enables us to approximate the fractional derivatives on irregular domain with high flexibility and good accuracy. Applying this local DQ formula in space and using weighted average (WA), predictor-corrector (PC) difference schemes in time, we then construct two linearized local RBFs-based DQ schemes for the two-dimensional nonlinear Riesz space-fractional advection-diffusion equations (ADEs). The validity of this local DQ formula is confirmed by numerical results. The feasibility and capability of both local RBFs-based DQ schemes are authenticated by the illustrative tests on some irregular domains and the numerical simulation of Gaussian pulse propagation.

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Acknowledgements

The authors express sincere thanks to the anonymous referees and editors for their valuable comments.

Funding

This research was supported by the Natural Science Foundation of Hunan Province of China (Nos. 2020JJ5514 and 2020JJ4554), and the Scientific Research Funds of Hunan Provincial Education Department (Nos. 19C1643, 19C1668 and 19B509).

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Correspondence to Xiaogang Zhu.

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Zhu, X., Li, J. & Zhang, Y. A local RBFs-based DQ approximation for Riesz fractional derivatives and its applications. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01183-w

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Keywords

  • Local DQ formula
  • Local RBFs-based DQ scheme
  • Space-fractional ADEs

Mathematics Subject Classification (2010)

  • 35R11
  • 65M70