As discussed in Steady-state 1-D Solution Methods , even though the Premixed Flame models are solved as steady-state problems, the steady-state solution algorithm sometimes requires pseudo time-stepping to condition the solution iterate. For this reason, we extend the discussion of the governing equations presented in 1-D Flame Equations to include the transient terms used in time-stepping procedures. The transient equations are obtained by adding the time derivatives to Equation 13–2 and Equation 13–3 , obtaining,

(13–34)

and

(13–35)

The full system now becomes a system of parabolic partial differential equations, rather than an ordinary differential equation boundary value system. Solution is obtained via the backward-Euler method. In this method, the time derivatives are approximated by finite differences as illustrated by

(13–36)

where the superscript indicates the time level and represents the size of the time step. All other terms are approximated with finite differences as before, but at time level . Since all variables are known at time level , the discretized problem is just a system of nonlinear algebraic equations for the dependent variable vector at time level .