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May 11, 2005, 05:25 
Advection equation....

#1 
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Hi all,
I have a question regarding a problem I am working on. I have a transport equation for a propogating surface (say the linear advection equation) but the surface moves due to its own propogation speed (say burning velocity) and also due to the flow field velocity. Now, if I write this equation I get d_phi/dt + u d_phi/dx + ub d_phi/dx = 0 where phi is a scalar variable, u is the flow velocity and ub is the burning velocity. Now, assume I am discretizing each term separately (i.e, I am not going to lump the two convective terms together) and if (say) u> 0 and ub< 0 in a 1D case, then should I use the sum of u+ub to determine the upwind direction for both convective terms (i.e, they have the same upqind direction) or should I use the respective velocity (u or ub) for each term (i.e, they will have different upwind directions in this case? This may seem like a trivial question, but I am not too sure of the answer! Thanks for any help. Any references to books would be appreciated. 

May 11, 2005, 13:42 
Re: Advection equation....

#2 
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Is there a specific reason why you are discretizing each convective term seperately?
My first thought is to decide the upwind cell from the sum of u and ub as both of them combined are responsible for effecting the value of phi at a cell. 

May 11, 2005, 17:33 
Re: Advection equation....

#3 
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Well, before any suggestion, I think it is important to know if u and ub are constants (independant of x).
If they are constants, then phi is simply the solution of the wave equation (completely dependant on the initial conditions), traveling at u+ub. If they are not constant, you have yourself a nonlinear wave equation which you can solve by various methods (mostly iterative) ... 

May 12, 2005, 05:32 
Re: Advection equation....

#4 
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u is the velocity field and is solved, so is not independent of x. ub is a function of phi, the scalar variable and therefore we have a nonlinear equation. However, it is linearized by making ub dependent on the old time step values. The reason for discretizing the terms separately is numerical (calculating wuth ub inside the Grad phi equation results in problems). So does the fact that one convective term is responsible for moving the wave to the left and the other for moving it to the right (for example) mean that I have a different upwind direction for each or should I use the upwind direction given by u+ub as suggested by Sachin?


May 12, 2005, 12:00 
Re: Advection equation....

#5 
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If you linearize the equation to a constant coefficient wave equation, it should not matter whether you discretize the terms separately or combine them.
Some easy tests: 1. In the discrete form, can you derive one form from the other by algebriac manipulations? 2. Do they give the same results when computed? 

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