The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I)

Abstract

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α∈(1,2]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order Oτ2+h4 for α∈(1,1.5), where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme.