Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula

doi:10.1007/s42967-021-00177-8

Abstract

Abstract: With the help of the asymptotic expansion for the classic L1 formula and based on the L1- type compact difference scheme, we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation. Three extrapolation formulas are presented, whose temporal convergence orders in L_∞-norm are proved to be 2, 3-α, and 4-2α, respectively, where 0 < α < 1. Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal 4-γ and 6-2γ order in L_∞-norm, respectively, where 1 < γ < 2. Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation, the fast extrapolation methods are obtained which reduce the computational complexity significantly while keeping the accuracy. Several numerical experiments confirm the theoretical results.

Key words: L1 formula, Asymptotic expansion, Fractional sub-diffusion equation, Fractional wave equation, Richardson extrapolation, Fast algorithm

CLC Number:

Ren-jun Qi, Zhi-zhong Sun. Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1313-1350.

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References

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