Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term

This paper aims to propose a high-order and accurate numerical scheme for the so- lution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we rst discretize the Riesz fractional derivative with a fourth-order nite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time compo- nents and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those pro- vided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an effcient algorithm for solving such problems and can arrive at the high-precision.