WENO schemes with Z-type non-linear weighting procedure for fractional differential equations

In this paper, a new fourth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. A set of Z-type non-linear weights is constructed based on the L_1 norm, yielding improved WENO scheme with more accurate resolution. The Caputo fractional derivative of order \alpha is split into a weakly singular integral and a classical second derivative. The classical Gauss-Jacobi quadrature is employed for solving the weakly singular integral. Also, a new WENO-type reconstruction methodology for approximating the second derivative is developed. Some benchmark examples are prepared to illustrate the efficiency, robustness, and good performance of this new finite difference WENO-Z scheme.

WENO schemes with Z-type non-linear weighting procedure for fractional differential equations Keywords:

finite difference scheme , Fractional differential equations , WENO-Z scheme