CFD Online Logo CFD Online URL
Home > Forums > Main CFD Forum

Time levels & averaging question

Register Blogs Members List Search Today's Posts Mark Forums Read

LinkBack Thread Tools Display Modes
Old   June 5, 2007, 12:16
Default Time levels & averaging question
Posts: n/a
Hello Folks, I was wondering if someone could please shed some light on this matter for me:

If I have a differential equation say:


and I wish to solve this equation using finite difference equations then what benefit is gained by averaging the u_t term spatially and u_x temporally?

In other words, I have seen the equation in finite difference form as:

0.5*[ (u(I,N+1)-u(I,N)) + (u(I-1,N+1)-u(I-1,N))]/dt = 0.5*[ (u(I,N)-u(I-1,N)) +(u(I,N+1)-u(I-1,N+1)) ]/dx

where 'I' denotes the spatial dimension and 'N' the time-level. I was wondering what benefit is gained through averaging one term spatially and the other temporally. I have been told it ensures that both sides of the equation are relating to the point (I+1/2, N+1/2) but I'm not sure what this actually achieves.

If someone could shed some light on this matter I'd be very grateful. Many thanks, F
  Reply With Quote

Old   June 5, 2007, 18:30
Default Re: Time levels & averaging question
Ananda Himansu
Posts: n/a
It gives you a scheme (which I remember as the Keller box scheme, though I could be wrong; check some text) which is second-order accurate in both space and time and yet has a very compact stencil. The order of accuracy of the discretization (and its consistency with the given differential eqn) can be checked by Taylor series expansion of the discretized eqn, term by term, about ANY point in space-time, though the point labeled (I-1/2,N+1/2) is most elegant for the discretization you show, as the cancellation of Taylor series terms is most obvious in that case. You could say that the discretization is "centered" about (I-1/2,N+1/2). The compactness of the stencil brings with it the not inconsiderable advantage that the number of boundary and initial conditions needed (just one of each) matches the natural number of bcs and ics required by the diff eqn, thus obviating the need for artificial boundary conditions.
  Reply With Quote


Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On

Similar Threads
Thread Thread Starter Forum Replies Last Post
Starting field averaging using libFunctionObject after certain time eelcovv OpenFOAM Programming & Development 25 December 7, 2015 23:28
Full pipe 3D using icoFoam cyberbrain OpenFOAM 4 March 16, 2011 10:20
export time steps for ensemble averaging Thomas Baumann Siemens 0 March 2, 2010 04:19
Transient simulation not converging skabilan OpenFOAM Running, Solving & CFD 12 September 17, 2007 17:48
unsteady calcs in FLUENT Sanjay Padhiar Main CFD Forum 1 March 31, 1999 12:32

All times are GMT -4. The time now is 03:54.