A method is proposed to augment the proper orthogonal decomposition basis functions with discontinuity modes to better capture moving discontinuities in reduced-order models. Moving discontinuities can be shocks in unsteady gas flows or bubbles in multiphase flow. The method is shown to work for a simple test problem using the first-order wave equation. A method for detecting discontinuities numerically is developed using mathematical morphology. This method is shown to properly identify the edges of bubbles in multiphase flow. (C) 2010 Elsevier B.V. All rights reserved.