Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.