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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 |

**User-agent**: |
KMail/1.5.4 |

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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,
Stephane.
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