How to implement transient equations with auxiliary terms

[HyungseonSong-plasma]

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Question

Hello, I'd like to develop the kernel with below equation forms.

$$\frac{\partial (Tn)}{\partial t} + \nabla\cdot k\nabla T$$

T,n are variables. My interest is how to implement $\frac{\partial (Tn)}{\partial t}$. In my understanding, I should use the two kernels modifying TimeDerivative with coupled variables, and kernel1 : $n\frac{\partial (T)}{\partial t}$ kernel2 : $T\frac{\partial (n)}{\partial t}$.
Are there any solutions for doing one kernel dealing with $\frac{\partial (Tn)}{\partial t}$?

[GiudGiud]

Hello

Which numerical method are you using? FE or FV?
And both n and T are variables?

You can create a TimeKernel derived class that does the n dTdt + T dndt.
It's a good first start. If you want to implement dTndt for conservation reasons, it's a little more complicated

[HyungseonSong-plasma]

Both n and T are varaibles and I am using FE. But when I tried n dTdt + T dndt, I usually got the there's no + operator message.
I see the CoupledTimeDerivative for T dndt first, it contains only about the v so I should modify Residual and OffDiagJacobian.
I'm not sure the below ones are correct.
Residual _test[_i][_qp] * _coef * _u[_qp] * _v_dot[_qp];
OffDiagJacobian _test[_i][_qp] * _phi[_j][_qp] * _coef * _u[_qp] * _dv_dot[_qp];

If above one is correct, then I should try
Residual _test[_i][_qp] * _coef * (_u[_qp] * _v_dot[_qp] + _v[_qp] * _u_dot[_qp]);
OffDiagJacobian _test[_i][_qp] * _phi[_j][_qp] * _coef *(_u[_qp] * _dv_dot[_qp] + _v[_qp] * _u_dot[_qp]);

[HyungseonSong-plasma]

Thanks GiudGiud, I can get the well compiled coupledcoeftimederivative following your suggestions dTdt + T dndt,
Last question is about your comments "If you want to implement dTndt for conservation reasons, it's a little more complicated".
If it's complicated, but is there a method for conservation form??. Because Example equations are actually to implement energy balance equation.

[GiudGiud]

The way to do a conservative form (so doing (Tn)(i+1) instead of T(i+1) n_(i+1) might be through a material property.

But maybe it s easier for you to redefine your equations using (T, Tn) as the variables instead of (T, n)

[GiudGiud]

Because Example equations are actually to implement energy balance equation.

My concern is only that, for a 1st order Euler scheme for example, the natural conservation of Tn would be achieved with

((Tn)_{t+1} - (Tn){t}) / dt

but when using the product rule on the derivative we end up with

n_{t+1} * (T_{t+1} - T_t}) / dt + T_{t+1} * (n_{t+1} - n_{t}) / dt 

which is not exactly the same. I think the first is more "conservative" but I am rusty on these things. It would be great to look into it.