I am not a mathematician and my question may not be well posed but I would appreciate any insight if you have it.
Suppose I have a model of a hydraulic system composed of nonlinear differential and algebraic equations.
The differential equations are discretized to yield purely algebraic equations, and the total equation system is then arranged in a lower triangular block form so that the equations can be solved sequentially from top to bottom.
If the model has algebraic loops, these will show as a rectangular block of coupled algebraic equations that do not fit in the lower triangular block form and which must be solved simultaneously using a nonlinear iterator like Newton-Raphson before proceeding.
But assume that I am using an implicit solver for the entire model anyway, so even the discretized differential equations will require a nonlinear iterator like Newton-Raphson if they are nonlinear.
For some reason the nonlinear algebraic loops seem to be especially challenging for the solver, and I suppose I am trying to understand why. My question is: what is the difference between solving N discretized nonlinear differential equations sequentially to some tolerance T, and solving a block of N coupled nonlinear algebraic equations to that same tolerance T (in both cases using an implicit method with a nonlinear iterator)?
Is the block of algebraic equations more challenging for the solver (as it appears to be)? If yes, can you say why, or give an example showing how the algebraic block might take more iterations or fail to converge altogether while a similar system of differential equations does not pose that problem?
Again, apologies if the question is ill-posed. If there is something I can do to make the question clearer, or if there is detail I should add, please let me know in the comments. Thank you!