Notes on the double nested Newton method

An sample 1d problem (heat diffusion with discontinuous heat conductivity)

This is a rewriting of my notes of Michael Dumbser class.

Casulli-Zanolli (CZ) method can be found in a recent paper [(Casulli 2012)]. It is needed to integrate equations of type:

\begin{equation} \frac{\partial Q}{\partial t}=\frac{\partial}{\partial x}\left(\lambda\frac{\partial T}{\partial t}\right)\\ \end{equation}

if \(\lambda\) presents some discontinuity for a value of t.

A generic \(\lambda\) function presenting a discontinuity in some point and being not-decreasing over the whole interval

The equation contains the derivative of this \(\lambda\) function and Newton’s method cannot converge in this case Considering this 1-d model, its explicit discretization can be:

\begin{equation} Q^{n+1}_{i}=Q^{n}+\frac{\Delta t}{\Delta x}\left(\lambda_{i+1/2}\frac{T^{n}_{i+1}-T^{n}_{i}}{\Delta x}-\lambda_{i-1/2}\frac{T^{n}_{i+1}-T^{n}_{i}}{\Delta x}\right)\\ \end{equation}

Its implicit discretisation

\begin{equation} Q^{n+1}_{i}=Q^{n}+\frac{\Delta t}{\Delta x}\left(\lambda_{i+1/2}\frac{T^{n+1}_{i+1}-T^{n+1}_{i}}{\Delta x}-\lambda_{i-1/2}\frac{T^{n+1}_{i+1}-T^{n+1}_{i}}{\Delta x}\right)\\ \end{equation}

However a workaround can be found by using a Jordan decomposition of the \(\lambda\) function into two non-decreasing functions.

\begin{equation} g(x)=g_{1}(x)-g_{2}(x)\\ \end{equation}
Here it is a representation of the differences between two functions that gives a spiky function