en>John of Reading: /* Different aspects of central differencing scheme */Typo fixing, replaced: spreaded → spread using AWB

2014-06-24T16:42:32Z

Different aspects of central differencing scheme: Typo fixing, replaced: spreaded → spread using AWB

Show changes

en>BD2412: /* Different aspects of central differencing scheme */Fixing links to disambiguation pages using AWB

2014-02-24T15:40:55Z

Different aspects of central differencing scheme: Fixing links to disambiguation pages using AWB

Show changes

en>BG19bot: WP:CHECKWIKI error fix for #62. Left offf http://. Do general fixes if a problem exists. -, replaced: www. → http://www. using AWB (9838)

2014-01-07T00:56:41Z

WP:CHECKWIKI error fix for #62. Left offf http://. Do general fixes if a problem exists. -, replaced: <ref>www. → <ref>http://www. using AWB (9838)

New page

{{intro-missing|date=December 2013}}
In case of transient problems, the simulations conducted using [[Computer-aided engineering|CAE]] packages require discretizing the governing equations both in space and time. Such problems are unsteady (for e.g.; flow problems) and therefore provide solution which varies with time for a particular position. Temporal discretization involves the [[Integral|integration]] of every term in different equations over a time step (Δt).
The spatial domain can be discretized to produce a semi-discrete form:<ref>{{cite web|url=http://aerojet.engr.ucdavis.edu/fluenthelp/html/ug/node993.htm |title=Spatial and Temporal Discretization}}</ref>

: <math>\frac{\partial \varphi}{\partial t}(x,t) = F(\varphi).~</math>

If the discretization is done using [[Backward differentiation formula|backward differences]];
The first order temporal discretization is given as:<ref name=" Selection of Spatial and Temporal discretization for Wetland modeling">[http://gwmftp.jacobs.com/Peer_Review/resolution.pdf Selection of Spatial and Temporal discretization]</ref>

: <math>\frac{\varphi^{n+1} - \varphi^{n}}{\Delta t} = F(\varphi),</math>

And the second order [[discretization]] is given as:

: <math>\frac{3\varphi^{n+1} - 4\varphi^{n}+\varphi^{n-1}}{2\Delta t} = F(\varphi),</math>

where

φ = a [[scalar (physics)|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(<math>\varphi</math>) is evaluated using implicit and explicit time integration.<ref>{{cite web|url=http://www.cfd-online.com/Wiki/Discretization_of_the_transient_term |title=Discretisation of transient term}}</ref>

== Description ==
The temporal discretization is done through [[Integral|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:

: <math>\frac{\varphi^{n+1} - \varphi^{n}}{\Delta t} = F( \varphi^{n+1} ),</math>

where f is weighing factor ranges between 0 and 1.
If
f = 0.0 results in the fully [[Explicit and implicit methods|explicit scheme]].
f = 1.0 results in the fully [[Explicit and implicit methods|implicit scheme]].
f = 0.5 results in the [[Crank–Nicolson method|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]]{{dn|date=December 2013}} terms.<ref>{{cite web|url=http://www.xmswiki.com/xms/GMS:Temporal_Discretization |title=Examples of Temporal Discretization}}</ref>

: <math> \int\limits_{t}^{t+\Delta t} F(\varphi) dt = [ f. F_\varphi^{t+\Delta t} + (1-f). F_\varphi^t ] \Delta t </math>

== Methods for evaluating function F(<math>\varphi</math>) ==
After discretizing the time derivative, function F(<math>\varphi</math>) remains to be evaluated. The function is now evaluated using implicit and explicit time integration.<ref name="Notes on Spatial and Temporal Discretization">[http://www.pc-progress.com/Documents/Notes_on_Spatial_and_Temporal_Discretization.pdf Jirka Simunek]</ref>

===Implicit Time Integration===
This methods evaluates the function F(<math>\varphi</math>) at a future time.

====Formulation====
The evaluation using Implicit Time Integration is given as:

: <math>\frac{\varphi^{n+1} - \varphi^{n}}{\Delta t} = F( \varphi^{n+1} ),</math>

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)):

: <math>\varphi^{n+1} = \varphi^{n} + \Delta t F( \varphi^{n+1} ),</math>

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(<math>\varphi</math>) at a current time.

====Formulation====
The evaluation using Explicit Time Integration is given as:

: <math>\frac{\varphi^{n+1} - \varphi^{n}}{\Delta t} = F(\varphi^n),</math>

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

: <math>\varphi^{n+1} = \varphi^{n} + \Delta t F( \varphi^n ),</math>

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==

{{reflist}}

[[Category:Computer-aided engineering]]

Central differencing scheme - Revision history

en>John of Reading: /* Different aspects of central differencing scheme */Typo fixing, replaced: spreaded → spread using AWB

en>BD2412: /* Different aspects of central differencing scheme */Fixing links to disambiguation pages using AWB

en>BG19bot: WP:CHECKWIKI error fix for #62. Left offf http://. Do general fixes if a problem exists. -, replaced: www. → http://www. using AWB (9838)