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18.3.2 Temporal Discretization

For transient simulations, the governing equations must be discretized in both space and time. 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 a time step $\Delta t$. The integration of the transient terms is straightforward, as shown below.

A generic expression for the time evolution of a variable $\phi$ is given by

 \frac{\partial\phi}{\partial t} = F(\phi) (18.3-15)

where the function $F$ incorporates any spatial discretization. If the time derivative is discretized using backward differences, the first-order accurate temporal discretization is given by

 \frac{\phi^{n+1} - \phi^{n}}{\Delta t} = F(\phi) (18.3-16)

and the second-order discretization is given by

 \frac{3\phi^{n+1} - 4\phi^{n} + \phi^{n-1}}{2\Delta t} = F(\phi) (18.3-17)


$\phi$ = a scalar quantity
$n+1$ = value at the next time level, $t + \Delta t$
$n$ = value at the current time level, $t$
$n-1$ = value at the previous time level, $t - \Delta t$

Once the time derivative has been discretized, a choice remains for evaluating $F(\phi)$: in particular, which time level values of $\phi$ should be used in evaluating $F$?

Implicit Time Integration

One method is to evaluate $F(\phi)$ at the future time level:

 \frac{\phi^{n+1} - \phi^{n}}{\Delta t} = F(\phi^{n+1}) (18.3-18)

This is referred to as "implicit'' integration since $\phi^{n+1}$ in a given cell is related to $\phi^{n+1}$ in neighboring cells through $F(\phi^{n+1})$:

 \phi^{n+1} = \phi^{n} + {\Delta t} F(\phi^{n+1}) (18.3-19)

This implicit equation can be solved iteratively at each time level before moving to the next time step.

The advantage of the fully implicit scheme is that it is unconditionally stable with respect to time step size.

Explicit Time Integration

A second method is available when the density-based explicit solver is used. This method evaluates $F(\phi)$ at the current time level:

 \frac{\phi^{n+1} - \phi^{n}}{\Delta t} = F(\phi^{n}) (18.3-20)

and is referred to as "explicit'' integration since $\phi^{n+1}$ can be expressed explicitly in terms of the existing solution values, $\phi^{n}$:

 \phi^{n+1} = \phi^{n} + {\Delta t} F(\phi^{n}) (18.3-21)

Here, the time step $\Delta t$ is restricted to the stability limit of the underlying solver (i.e., a time step is limited by the Courant-Friedrich-Lewy condition). In order to be time-accurate, all cells in the domain must use the same time step. For stability, this time step must be the minimum of all the local time steps in the domain. This method is also referred to as "global time stepping''.

The use of explicit time stepping is fairly restrictive. It is used primarily to capture the transient behavior of moving waves, such as shocks, because it is more accurate and less expensive than the implicit time stepping methods in such cases. You cannot use explicit time stepping in the following cases:

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