Missing from the Original Fortran Codes

[gabrielgellner]

The code from ODE.jl is clearly based on the Hairer and Wanner monograph, but it is not clear at times how it has been changed. Things that need to be investigated:

• Some of the code has some magical constants like timeout_const = 5 which feel unmotivated in the current form.
• It is not clear that the stiffness check has been ported over.

Features that are missing that are known include:

• [] the Hermite interpolation is of the incorrect order
  • [X] Dopri5 should have 4th order
  • [] Dop853 should have 7th order
  • [X] Really there are special forms derived by Shampine for the forms of these Hermite coefficients that take advantage of the ode formulation. These should simply be ported over from the original codes.
• [] Event detection.