replace residual-based convergence criteria with Shampine algorithm

[BenWibking]

Describe the proposal
We use a variety of residual-based convergence criteria for the Newton-Raphson solver in the radiation code. This has failed in problems where the residual is badly scaled.

There is a semi-rigorous way to decide convergence that bounds the distance from the true solution described in detail in section 3 of Shampine (1980): https://epubs.siam.org/doi/10.1137/0901005.

If you know both the contraction factor $r$ and the difference in successive iterates, you can bound the difference of the current guess $y^{m+1}$ from the exact solution $y^{\star}$ (see Equation 13):

The naive estimate of the contraction factor $r$ is that available from the most recent iterates $r_m$:

This provides a lower bound on $r$. However, the problem is that a mathematically rigorous upper bound on the contraction factor $r$ is not computable from the iterates (that requires bounds on the norm of the derivative of the function or a Lipschitz constant [1][2][3][4]). ("In general, we must anticipate the possibility that $r_m$ is is a misleading estimate for $r$ and in particular, could be too small.")

Nonetheless, Shampine advocates that a reasonably robust estimate of $r$ can be computed, if you do so carefully: "It is far safer to presume a contraction with rate $r$ from the initial point $y_0$ and to use the largest observed $r_m$ as the best estimate for $r$ available." This appears to be worth trying.

Describe alternatives you've considered
Keep existing ad-hoc criteria.

Additional context
Convergence often fails when the ratio of the heat capacity of radiation and matter is $\gtrsim 15$ orders of magnitude. In this case, we have found that the residual is badly scaled. However, this may also indicate a numerical precision limitation.

There are theorems for exact error bounds, but they require bounds on the derivative of $f$ (or a Lipschitz constant) [1][2][3][4].

[1] https://en.wikipedia.org/wiki/Kantorovich_theorem
[2] https://link.springer.com/article/10.1007/BF01463998
[3] https://epubs.siam.org/doi/10.1137/0711002
[4] https://link.springer.com/article/10.1007/BF01389624

cc @chongchonghe @markkrumholz