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 α∈(1, 2) and the error is shown to have the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time, where τ denotes the step size and dl, l=3, 4, ⋯ and dl∗, 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(τ3-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(τ4-α) and O(τ2(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.