In this study, one explicit and one implicit finite differencescheme is introduced for the numerical solution of time-fractional Rieszspace diffusion equation. The time derivative is approximated by the standard Gr¨unwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four.The stability and convergence of the proposed methods are studied. Itis proved that the implicit scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustratethe efficiency and accuracy of the proposed methods. Numerical resultsconfirm that the accuracy of present schemes is of order one. Manuscript profile