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Discretization of the transient term

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To account for transient effects, the governing equations must be discretized in time. As it turns out, the temporal discretization is slightly easier to deal with than that for the spatial effects. Since the governing equation is hyperbolic/parabolic in time, the solution at time t depends upon its history and not on its future. Transient effects are usually dealt with by using a time stepping procedure, with an initial condition provided. The time dimension is divided into a set of discrete time steps, each of size \Delta t \,. The solution algorithm therefore marches forward in time, computing a solution at each time step. The spatial discretization for the time-dependent equations is identical to the steady-state case.
Temporal discretization involves the integration of every term in the differential equations over the time step \Delta t \,. The integration of transient effects takes several forms, each yielding a different accuracy. For simplicity, we express the time dependent transport of a scalar  \phi \, as
 \frac {\partial (\rho \phi)} {\partial t} = l(\phi)
where the function  l \, is an operator that incorporates all of the non-transient terms, namely: diffusion, convection, and source terms.
Integrating the above equation over a control volume yields
 \int_{V} \frac {\partial (\rho \phi)} {\partial t} dV= \int_{V} l(\phi) dV
After the spatial dicretization has been performed using the techniques describes in the other sections, we obtain
 \frac {\partial (\rho \phi)} {\partial t} V= L(\phi)
where L denotes the spatial discretization operator, and V denotes the volume.

(to be continued)

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