Quantifying the degree of nonlinearity in a heat transfer problem

Question

I’m working with a heat equation of the form. $$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal conductivity $k(T)$. Ordinarily this would lead to a system of nonlinear equations that would need to be solved. I was searching for any other way to solve this problem without using a nonlinear solve at every timestep.

I found an approach in Reddy & Garling’s The Finite Element Method for Heat Transfer and Fluid Dynamics. In it, they take a quasi-linearization approach which attempts to predict the temperature T* at the current timestep ${n+1}$ using values from previous time steps $T_n$, $T_{n-1}$ using the expression:

$$T^*=\frac{3}{2}T_n-\frac{1}{2}T_{n-1}$$

So then, for any timestepping scheme used to discretize the equation, one can evaluate the coefficients of the equation using $T^*$; thus linearizing the equation.

The authors don’t give a rigorous explanation of when this technique is applicable, only stating that it is appropriate for “mildly nonlinear problems”. I’m not sure how nonlinearity is quantified in general, much less for heat transfer problems. I tried searching for literature on this technique, but it is hard to find since the authors don’t really give it a name. It appears to be some sort of finite difference extrapolation using a 3 point interpolation. If so, i would think that it would be applicable for coefficients that are roughly linear or quadratic in temperature. Is there a way i can justify this rigorously? If not, how can i quantify “mildly nonlinear” in the context of this problem?

Answer 1

A quick and dirty approach to quantify the nonlinearity could be to evaluate

$$\frac{1}{k}\frac{dk}{dT}\frac{\partial T}{\partial t}\delta t.$$

It's dimensionless and quantifies how much relative change you can expect in $k$ over a single timestep $\delta t$. The rough idea being that, if the thermal conductivity exhibits a large relative sensitivity to temperature and the temperature is changing a lot in a single timestep, then the error you incur by freezing the coefficients at $T_n$ in order to evaluate $T_{n + 1}$ could be large. By using the extrapolation method, you might be able to effectively increase the order of accuracy -- it's like assuming that the temperature is piecewise linear in time within each timestep rather than piecewise constant. You can always make the problem mildly nonlinear by reducing the timestep enough, but with the extrapolation you might be able to use larger timesteps.

I don't know the word for this trick either, which is bad because I've used it. You might be able to Taylor expand like crazy and show that the extrapolation scheme + Crank-Nicholson is formally 2nd-order accurate, even without doing a fully implicit discretization in time. I'm guessing this has already been done to study stability of methods that use a discontinuous Galerkin discretization in time.