► A new numerical model of the nonlinear diffusive shock acceleration is presented. ► We perfomed the modeling of particle acceleration in supernova remnants (SNRs). ► The forward and reverse shocks (RS) are included in the consideration. ► The spectra of cosmic rays (CRs) released into ISM from a SNR are determined. ► The role of the RS in the production of CR ions and positrons is discussed. A new numerical model of the nonlinear diffusive shock acceleration is presented. It is used for modeling of particle acceleration in supernova remnants. The model contains coupled spherically symmetric hydrodynamic equations and the transport equations for energetic protons, ions and electrons. The forward and reverse shocks are included in the consideration. The spectra of cosmic rays released into interstellar medium from a supernova remnant are determined. The role of the reverse shock in the production of cosmic ray ions and positrons is discussed.