# Domain of Dependence of 1D First-order Linear Convection

 Register Blogs Members List Search Today's Posts Mark Forums Read January 10, 2018, 23:37 Domain of Dependence of 1D First-order Linear Convection #1 New Member   abcd efgh ijkl Join Date: Oct 2015 Posts: 26 Rep Power: 10 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   January 11, 2018, 03:25 #2
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Filippo Maria Denaro
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 Originally Posted by alibaig1991 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.  Tags mathematical analysis, pde Thread Tools Search this Thread Show Printable Version Email this Page Search this Thread: Advanced Search Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode 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 OffTrackbacks are Off Pingbacks are On Refbacks are On Forum Rules Similar Threads Thread Thread Starter Forum Replies Last Post Ethan_Sparkle CFX 41 June 14, 2017 07:22 hsezsz CFX 2 October 13, 2016 06:58 gerritgroot OpenFOAM Running, Solving & CFD 0 September 30, 2015 20:06 immortality OpenFOAM Pre-Processing 11 August 21, 2013 08:45 Kiat110616 CFX 4 April 3, 2011 22:43

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