In this work, we generalize the Compositional Space Parameterization (CSP) approach, which was originally developed for compositional two-phase reservoir flow simulation. Tie-line based parameterization methods [1]–[3] were motivated by insights obtained from MOC (Method of Characteristics) theory. The MOC based analytical theory [4] has provided deep understanding of the interactions between thermodynamics and flow. In our adaptive framework, tie-lines are used to represent the solution route of multi-component multiphase displacements. The tie-line information is used as a preconditioner for EOS computations in general-purpose compositional flow simulation.