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Re: [ff3d-users] Could you help me confirm a time-dependent PDE?

From: Stephane Del Pino
Subject: Re: [ff3d-users] Could you help me confirm a time-dependent PDE?
Date: Wed, 25 Feb 2004 17:50:43 +0100
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On Wednesday 25 February 2004 00:20, Robert Li wrote:
> Dear Stephane Del Pino,
>   I tested ff3d with a realistic thermal PDE. However,
> I found that the result of ff3d was different from
> that of SVHEAT and TEMP/W, which are also FEM solution
> software. But the other two have close result, or
> almost same.
>   I am not sure about how to write PDE in input file.
> For example, It seems that -div((f2)*grad(T)) = 0; and
> -div((f2)*grad(T)) = 0; are different. But I don't
> know why?

You probably made a mistake writing the mail, the two formulae are the 
same ... Could you give more details, please?

>  Could you help me confirm the following
> time-dependent PDE?
> 1.Real Heat flow Equation
> dx(Kx*dx(T)) + dy(Ky*dy(T)) +
> dz(Kz*dz(T))=(Cp+Lf*Theat*Theatu1)*(dt(T))
> where, T=Temperature,
> dt=time
> 2.I expressed it in ff3d input file as follows:
> T-div((f2)*grad(T)) = Tn;
I think that this last equation is not equivalent to the first one.
Let me explain which mistake I think you did:

you want to solve something like
        a(x)*dt(T)-div(grad(T)) = 0
Using an Euler implicite scheme, the time discretization is:
        a(x)*(T(n+1)-T(n))/Dt - div(grad(T(n+1))) = 0
The this is *not* equivalent to
        T(n+1)-div(Dt/a(x)*grad(T(n+1))) = T(n)
Since a(x) cannot go inside the divergence term. A correct way of doing it 
could be:
        a(x)*T(n+1) - div(Dt*grad(T(n+1))) = a(x)*T(n)

Please tell me if that helps.

Best regards,
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