Unconditionally Stable and Convergent Difference Scheme for Superdiffusion with Extrapolation

Abstract

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order $$\alpha \in (1, 2)$$ α ∈ ( 1 , 2 ) and the error is shown to have the asymptotic expansion $$ \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{5-\alpha } + \cdots \big ) + \big ( d_{2}^{*} \tau ^{4} + d_{3}^{*} \tau ^{6} + d_{4}^{*} \tau ^{8} + \cdots \big ) $$ ( d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯ ) + ( d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯ ) at any fixed time, where $$\tau $$ τ denotes the step size and $$d_{l}, l=3, 4, \dots $$ d l , l = 3 , 4 , ⋯ and $$d_{l}^{*}, l\,=\,2, 3, \dots $$ d l ∗ , l = 2 , 3 , ⋯ are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order $$O (\tau ^{3- \alpha }), \alpha \in (1, 2)$$ O ( τ 3 - α ) , α ∈ ( 1 , 2 ) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are $$O ( \tau ^{4- \alpha })$$ O ( τ 4 - α ) and $$O ( \tau ^{2(3- \alpha )}), \alpha \in (1, 2)$$ O ( τ 2 ( 3 - α ) ) , α ∈ ( 1 , 2 ) , respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.