A new computing explicit technique RDSM is described for solution of one-dimensional nonsteady fluid flow problems in multilayer systems; homogeneous materials or two-phase mixtures are calculated with taking into account velocity and temperature nonequilibrium of components. Numerical method of integration of mathematical model system of equations is presented. An assumption of equal pressures is taken as a condition of common deformation of mixture components. Analytic solutions, constructed within equations of heterogeneous media mechanics, have been used to test the method. The method describes singularities of Shockwave processes in multiphase mixtures with good accuracy; it describes multiwave structures, separation, behaviour of components mass concentrations.