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

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 $L_\infty$-norm are proved to be 2, $3-\alpha$, and $4-2\alpha$, respectively, where $0<\alpha <1$. Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal $4-\gamma$ and $6-2\gamma$ order in $L_\infty$-norm, respectively, where $1<\gamma <2$. 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.

Funding

National Natural Science Foundation of China(11671081)