A novel numerical scheme for the simulation of viscoelastic fluid flows based on the lattice Boltzmann method is presented over a large range of Weissenberg numbers. In this method, two types of distribution functions are defined, one is scalar density for the evolution of the momentum and the other is tensor density for the stress tensor evolution. Through treating the source term properly, the redundant term lacking of clear physical meaning in the recovered stress tensor equation can be removed. The numerical results for the two-dimensional channel flow are found to be in good agreement with analytical results, which shows that the present method for solving viscoelastic fluid has a good accuracy. And as a stringent test, an Oldroyd-B fluid in a lid-driven cavity is simulated. The calculated results further demonstrate that the lattice Boltzmann method proposed in this paper is stable at large values of the Weissenberg numbers and does indeed perform well to capture the structural patterns of elastic instabilities. (C) 2012 Elsevier B.V. All rights reserved.