We present a set of algorithms for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a compressible two-phase system. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the timestep. In the absence of gravity and capillary forces, the resulting nonlinear system of discrete transport equations can be permuted to lower triangular form with a simple topological-sorting method from graph theory. This gives an optimal nonlinear solver that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of nonlinear scalar equations that can be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in run times and increases the feasible timestep sizes. In fact, one can prove that the solver is unconditionally stable and will produce a solution for arbitrarily large timesteps. For cases with gravity, the same method can be applied as part of a nonlinear Gauss–Seidel method. Altogether, our results demonstrate that sequential splitting combined with single-point upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.