Temporal discretization

In the fields of applied physics and engineering, transient problems are often solved by conducting simulations using computer-aided engineering (CAE) packages, which require discretizing the governing equations in both space and time. Such problems are unsteady (e.g. flow problems), and therefore require solutions in which position varies as a function of time. Temporal discretization involves the integration of every term in different equations over a time step (Δt). The spatial domain can be discretized to produce a semi-discrete form:^[1]

${\frac {\partial \varphi }{\partial t}}(x,t)=F(\varphi ).~$

If the discretization is done using backward differences; The first order temporal discretization is given as:^[2]

${\frac {\varphi ^{{n+1}}-\varphi ^{{n}}}{\Delta t}}=F(\varphi ),$

And the second order discretization is given as:

${\frac {3\varphi ^{{n+1}}-4\varphi ^{{n}}+\varphi ^{{n-1}}}{2\Delta t}}=F(\varphi ),$

where

    φ   = a scalar quantity.

    n+1 = value at the next time level,t+Δt.

    n   = value at the current time level,t.

    n-1 = value at the previous time level, t-Δt.

The function F($\varphi$) is evaluated using implicit and explicit time integration.^[3]

Description

The temporal discretization is done through integration over time on the general discretized equation. First, values at a given control volume P at time interval t are assumed and then value at time interval t+Δt is found. This method states that the time integral of a given variable is equal to a weighted average between current and future values. The integral form of the equation can be written as:

${\frac {\varphi ^{{n+1}}-\varphi ^{{n}}}{\Delta t}}=f.F(\varphi ^{{n+1}})+(1-f).F(\varphi ^{{n}}),$

where f is weighing factor ranges between 0 and 1. If

  f = 0.0 results in the fully explicit scheme.
  f = 1.0 results in the fully implicit scheme.
  f = 0.5 results in the Crank-Nicolson scheme.
                                  

For any control volume this integration holds true for any discretized variable. The following equation is obtained when applied to the governing equation including full discretized diffusion, convection, and source terms.^[4]

$\int \limits _{{t}}^{{t+\Delta t}}F(\varphi )dt=[f.F_{\varphi }^{{t+\Delta t}}+(1-f).F_{\varphi }^{t}]\Delta t$

Methods for evaluating function F($\varphi$)

After discretizing the time derivative, function F($\varphi$) remains to be evaluated. The function is now evaluated using implicit and explicit time integration.^[5]

Implicit Time Integration

This methods evaluates the function F($\varphi$) at a future time.

Formulation

The evaluation using Implicit Time Integration is given as:

${\frac {\varphi ^{{n+1}}-\varphi ^{{n}}}{\Delta t}}=F(\varphi ^{{n+1}}),$

This is called implicit integration as φ(n+1) in a given cell is related to φ(n+1) in neighboring cells through F(φ(n+1)):

$\varphi ^{{n+1}}=\varphi ^{{n}}+\Delta tF(\varphi ^{{n+1}}),$

In case of Implicit method, the setup is unconditionally stable and can is handle large time step (Δt). But, stability doesn't mean accuracy. Therefore, large Δt affects accuracy and defines time resolution. But, behavior may involve physical timescale that needs to be resolved.

Explicit Time Integration

This methods evaluates the function F($\varphi$) at a current time.

Formulation

The evaluation using Explicit Time Integration is given as:

${\frac {\varphi ^{{n+1}}-\varphi ^{{n}}}{\Delta t}}=F(\varphi ^{n}),$

And is referred as explicit integration since φ(n+1) can be expressed explicitly in the existing solution values, φ(n):

$\varphi ^{{n+1}}=\varphi ^{{n}}+\Delta tF(\varphi ^{n}),$

Here, the time step (Δt) is restricted by stability limit of the solver (i.e., time step is limited by the Courant–Friedrichs–Lewy condition. To be accurate w.r.t time, same time step should be used in all the domain and to be stable, the 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".

Examples

Many schemes use Explicit Time Integration. Some of these are as follows :

• Von Neumann stability analysis.
• Courant–Friedrichs–Lewy condition.
• Lax–Wendroff method.
• Runge-Kutta Method.

See also

• Finite element method
• Explicit and implicit methods
• Chi-Wang Shu

References

1. "Spatial and Temporal Discretization".
2. Selection of Spatial and Temporal discretization
3. "Discretisation of transient term".
4. "Examples of Temporal Discretization".
5. Jirka Simunek