A mathematical model is presented to simulate water and solute transport in a highly viscous fluid with a water-permeable, elastic immersed membrane. In this model, fluid motion is described by Stokes flow, whereas water fluxes across the membrane are driven by transmural pressure and solute concentration differences. The elastic forces, arising from the membrane being distorted from its relaxed configuration, and the transmembrane water fluxes introduce into model solutions discontinuities across the membrane. Such discontinuities are faithfully captured using a second-order explicit jump method [A. Mayo, SIAM J. Numer. Anal., 21 (1984), pp. 285-299], in which jumps in the solution and its derivatives are incorporated into a finite-difference scheme. Numerical results suggest that the method exhibits desirable volume accuracy and mass conservation. © 2006 Society for Industrial and Applied Mathematics.