We present a general framework for the computation of the thermodynamic behavior of multi-phase, multi-component systems. In this framework we use the fact that for given values of pressure and temperature the compositional space can be represented as a high dimensional simplex, while the phase equilibrium conditions can be represented by a low dimensional tie-simplex. We parameterize the high dimensional compositional space using a lower dimensional tie-simplex subspace. The tie-simplex Computation and interpolation procedures complement the parameterization and complete the mathematical framework. We demonstrate that the tie-simplex space fully describes the Solution route for displacements involving multi-phase mixtures. The efficiency of the method is demonstrated for several multi-phase equilibrium problems of practical interest. The tie-simplex parameterization can be performed in a preprocessing step to replace the flash computations completely, or can be used as an initial guess to accelerate the convergence of standard procedures. For general-purpose flow simulation, adaptive construction of the necessary tie-simplexes during the course of a simulation is an efficient strategy. (C) 2009 Elsevier B.V. All rights reserved.