Markovian outgoing wave boundary conditions are introduced as an approximate method to reduce the size of the computational grid for time integration of the time-dependent Schrödinger equation. The ratio and polynomial methods developed as open boundary conditions are applied to the wave function at the boundaries of the computational grid. This computational method is used to study the wave packet dynamics for a metastable well, a double well, and strong-field ionization of a model atom. Accurate results demonstrate that this method can significantly reduce the number of grid points required in a dynamical calculation for quantum dynamical problems. © 2012 Wiley Periodicals, Inc.