Needs for interfacing with partitioned DAE integrator

Note: This is a draft that will evolve as the numerical methods and software are developed.

With the mathematical model

GridKit will need provide the SUNDIALS integrator with the $\tilde{\mathbf{f}}^{{r}}$ functions and Jacobians (although this might change to a mass matrix style interface). I expect these to have essentially the same interface as IDA's IDAResFn and IDALsJacFn. Instead of having one of each like IDA does currently, there will need to be $N$. The dimensions and content of the N_Vectors passed to the residual function for a component is very important. I'm tentatively thinking that the state y will be a ManyVector representing $[\mathbf{y}^{{r}}, \mathbf{w}^{{r}}]$. A ModelEvaluator will need some way to take in the coupling state $\mathbf{w}^{{r}}$. Perhaps this can be done through the parameters. I think standard N_Vectors of dimension $n_r$ can be used for yp and res in the residual function. The biggest feature needed here is some way to partition a ModelEvaluator into the $N$ components. The partitioned DAE integrator class will need to be passed something like a std::vector of ModelEvaluator or a composite object like SystemModel where the residual function and other properties can be indexed by component.

The implementation of the residual function for $h$ (sum over $\mathbf{h}^{{b,r}}$) in GridKit will need to leverage the special summation structure and potential for parallelism. The continuous partitioned Euler method performs an algebraic solve involving not just $h$ but also algebraic constraints from the components (denoted by $\mathbf{g}^{{r}}$ in report). A ModelEvaluator will need to provide this $\mathbf{g}^{{r}}$ function. Another option is to pass SUNDIALS one monolithic residual function that concatenates the $h$ and $g$ functions. Both will require some extra bookkeeping to indicate which variables are differential and which are algebraic. Ideally a ModelEvaluator will use a convention of putting differential variables first followed by algebraic without interweaving them. An index to the offset of the algebraic variables within a ModelEvaluator is needed. I suspect this interface will evolve significantly as we settle on the nonlinear solver and usage of tricks like nonlinear elimination.