Meshless methods for convection-dominated flow problems have the potential to reduce the computational effort required for a given order of solution accuracy compared to mesh-based methods. The state of the art in this field is more advanced for elliptic partial differential equations than for time-dependent convection diffusion problems. In this paper, we introduce a new meshless method that it based on combining the modified method of characteristics with the radial basis functions during the solution reconstruction. The method belongs to a class of fractional time-stepping schemes in which a predictor stage is used for the discretization of convection terms and a corrector stage is used for the treatment of diffusion terms. Special attention is given to the application of this method to solve convection-dominated flow problems in two-dimensional domains. Numerical results are shown for several test examples including the incompressible Navier Stokes equations and the computed results support our expectations for a stable and highly accurate meshless method. (C) 2016 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.