The numerical solution of a hyperbolic or a convection-dominated parabolic partial differential equation is challenging due to the large local gradients that are present in the solution. A possible method to track the sharp fronts that are associated with large gradients is to adapt the grid and this can be done dynamically or statically. i.e. at discrete points of time during the simulation. In this paper, a novel approach that is based on combining the high-order WENO scheme with a static moving grid method is presented. The proposed algorithm is tested on the viscid Burgers' equation, the linear advection equation and the population balance equation that describes particle growth in emulsion polymerization. Enhancements in the performance are observed in all case studies when compared with the conventional WENO scheme on a uniform grid making it a promising alternative when dealing with similar problems. (C) 2009 Elsevier Ltd. All rights reserved.