CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Domain of Dependence of 1D First-order Linear Convection

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

Like Tree1Likes
  • 1 Post By FMDenaro

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   January 11, 2018, 00:37
Default Domain of Dependence of 1D First-order Linear Convection
  #1
New Member
 
abcd efgh ijkl
Join Date: Oct 2015
Posts: 26
Rep Power: 11
alibaig1991 is on a distinguished road
I know about the domain of dependence of 2nd order hyperbolic equations. It is defined using characteristics and initial/boundary data. But how do we define domain of dependence for 1st order hyperbolic equation? There will be only one characteristic.
Thanks
alibaig1991 is offline   Reply With Quote

Old   January 11, 2018, 04:25
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,893
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by alibaig1991 View Post
I know about the domain of dependence of 2nd order hyperbolic equations. It is defined using characteristics and initial/boundary data. But how do we define domain of dependence for 1st order hyperbolic equation? There will be only one characteristic.
Thanks

Well, a first order PDE is classified by definition only hyperbolic. As example for a 2D case, being a and b real functions of (x1,x2).

a*df/dx1+b*df/dx2=0 -> df/dx1+(b/a)*df/dx2=0

so that you can make at system with the total differential

df = dx1*df/dx1+dx2*df/dx2 =dx1*(df/dx1+dx2/dx1*df/dx2)

and see that on the curves of the plane (x1,x2) dx2/dx1=b/a you get df=0.
A classical example is x1=t, x2=x, a=1, b=u=const.
The curve dx/dt=u is a straight line in the (x,t) plane. That means the function at a point Xp and at a time T is equal to the value of the initial data f(Xp-u*T,0). Thus, you can trace the domain of dependence at the time t=0.
alibaig1991 likes this.
FMDenaro is offline   Reply With Quote

Reply

Tags
mathematical analysis, pde

Thread Tools Search this Thread
Search this Thread:

Advanced Search
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 Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Can CFX model periodic heat transfer problem Ethan_Sparkle CFX 41 June 14, 2017 08:22
Sudden increase the residual of Maxwell's equations hsezsz CFX 2 October 13, 2016 07:58
First order in fvSchemes does not seem to work gerritgroot OpenFOAM Running, Solving & CFD 0 September 30, 2015 21:06
error on turbulence scheme immortality OpenFOAM Pre-Processing 11 August 21, 2013 09:45
CFX domain comparison Kiat110616 CFX 4 April 3, 2011 23:43


All times are GMT -4. The time now is 13:27.