High Order Finite Difference WENO Schemes For Fractional Differential Equations

Recently, numerical methods of fractional differential equations have been de-veloped rapidly. However, the case with discontinuous initial conditions is paid less attention, and weighted essentially non-oscillatory (WENO) schemes could maintain nonoscillatory in numerical solution of hyperbolic partial differential equations and nonlinear degenerate parabolic equations. Therefor, this thesis is the first to advance high order finite difference WENO schemes for fractional differential equations.On the one hand, we discuss the stability of sixth-order weighted essentially non-oscillatory (WEN06) schemes which is proposed by Liu et al.,[28]. Fourier transform and Taylor formula are adopted to expand WEN06. The CFL numeber is estimated by numerical experience, and WEN06is stable coupled with forward Euler (FE) integra-tion method or explicit Runge-Kutta (ERK) methods.Further, a(1<α≤2) order Caputo’s fractional derivative is splitted into second-order derivative and weakly singular integral. Firstly, second-order derivative is dis-cretized by WENO6in order to get higher-order accuracy. Then based on the Jacobi orthogonal polynomials, weakly singular integral is exactly calculated. So fractional derivative is approximated in high order of accuracy. Finally, numerical results demon-strate the effectiveness of WENO schemes.