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A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations

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Abstract

In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order \(\alpha \) \((1<\alpha \le 2)\) into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

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Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 210016, People’s Republic of China

    Yan Zhang & Jun Zhu

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Weihua Deng

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  1. Yan Zhang
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  2. Weihua Deng
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  3. Jun Zhu
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Correspondence to Jun Zhu.

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Yan Zhang: Research was supported by Science Challenge Project, No. TZ2016002

Weihua Deng: Research was supported by NSFC grant 12071195 and the AI and Big Data Funds under Grant No. 2019620005000775

Jun Zhu: Research was supported by NSFC grant 11872210 and Science Challenge Project, No. TZ2016002.

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Zhang, Y., Deng, W. & Zhu, J. A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations. J Sci Comput 87, 73 (2021). https://doi.org/10.1007/s10915-021-01486-z

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