Temporal Discretization: The Transient Term

Abstract

The discussions in previous chapters assumed steady state conditions, which did not require the discretization of the transient term. Accounting for transient phenomena adds a new dimension to the problem. However since transient variations are parabolic by nature, there is no need to define a field in the time dimension, as is the case for the spatial domain. In general only one or two additional variable fields, or time levels, are stored (depending on the numerical order of the selected scheme). Another difference with steady state configurations is that transient systems are modeled using a time stepping procedure. Starting with an initial condition at time $t = t_{0}$, the solution algorithm marches forward and finds a solution at time $t_{1} = t_{0} +\Delta t_{1}$. The solution found is the initial condition for the next time step and is used to obtain the solution at time $t_{2} = t_{1} +\Delta t_{2}$. The process is repeated until the required time is reached. The focus of this chapter is on techniques used for the discretization of the transient term. Two approaches for developing transient schemes are presented. In the first one Taylor expansions are used to express the transient term with the aid of nodal values. This is in effect a finite difference discretization. In the second approach the finite volume method is used on a pseudo time element in a similar fashion to what was done to the convection term. Several transient schemes are presented and their characteristics discussed.