A novel method for integrating the time-dependent Schrodinger equation is presented. Hydrodynamic quantum trajectories are used to adaptively define the boundaries and boundary conditions of a fixed grid. The result is a significant reduction in the number of grid points needed to perform accurate calculations. The Eckart barrier, along with uphill and downhill ramp potentials, was used to evaluate the method. Excellent agreement with fixed boundary grids was obtained for each example. By moving only the boundary points, stability was increased to the level of the full fixed grid. (c) 2006 Elsevier B.V. All rights reserved.