For steady-state problems, the 0-D homogeneous systems described in Homogeneous 0-D Reactor Models employ Reaction Design’s numerical solver, Twopnt, to determine a solution to the set of algebraic equations governing these Reactor Models. Twopnt solves the system of algebraic equations by first applying a damped modified Newton algorithm to the set of nonlinear algebraic equations represented by the steady-state versions of Equation 8–1 , Equation 8–9 , Equation 8–19 , Equation 8–20 , Equation 8–21 , Equation 8–147 , and Equation 8–141 . However, in the event that the Newton algorithm fails to converge during the iteration, the solution estimate will automatically be conditioned by integration of the time-dependent version of the equations over a fixed number of time steps. This time-stepping procedure provides a new starting estimate for the Newton algorithm that is closer to the steady-state solution, increasing the likelihood of convergence for the Newton method. After time-stepping, the Newton algorithm tries again for convergence and, if this fails, Twopnt takes additional time steps on the transient solution to further improve the initial iterate. Ultimately, the Newton iteration converges on the steady-state solution.