added method ROK4a

[cwittens]

Checklist

• [x] Appropriate tests were added
• [ ] Any code changes were done in a way that does not break public API
• [ ] All documentation related to code changes were updated
• [ ] The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• [ ] Any new documentation only uses public API

Additional context

added method ROK4a from Rosenbrock--Krylov Methods for Large Systems of Differential Equations I dont think OrdinaryDiffEq currently utilizes Krylov Solvers for solving the Jacobian, but I need the _transformtab(Alpha,Gamma,B,Bhat) in function ROK4aTableau() for a different project, so I thought I might as well implement the method, if I already added the Tableau.

[ChrisRackauckas]

I dont think OrdinaryDiffEq currently utilizes Krylov Solvers for solving the Jacobian

That's a choice. Just change linsolve = KrylovJL_GMRES() and it'll use a Krylov method. The tableau and the way the linear equations are solved are not 1-1.

[cwittens]

Ohh I didn't know that. What do you mean with The tableau and the way the linear equations are solved are not 1-1. ?

[ChrisRackauckas]

Every tableau can be used with every linear solver, and every linear solver can be used with every tableau.

[ChrisRackauckas]

What's the next steps here?

[cwittens]

are the failed checks from this method? I thought they were general. Otherwise I will try and look for a fix.

The "different project" I was talking about is something in Fortran where I needed the coefficients in the a,C,b,btilde,d,cformat and not Alpha,Gamma,B,Bhat as given in the paper, which is why I used _transformtab(Alpha,Gamma,B,Bhat). And after already implementing the Alpha,Gamma,B,Bhat coefficients, i thought I made as well add this as a new method, but I didnt plan to do more with it in julia/ordinarydiffeq.

[ChrisRackauckas]

The failed checks were from the method, but it was just a typo so I handled it.

Thanks! Did you benchmark it on a few problems to see how it works in practice?

[cwittens]

Thanks. Nop, I only checked if it works but did not benchmark it in any way.