Matrix-free iteration methods and implicit ODE solvers

[percolator]

Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we only need the driving function to advance to the next timestep. However, when using an implicit method, a non-linear system of equations needs to be solved (f(x) = 0), where typically an iterative quasi-Newtonian method is used to find the solution when the derivative of f(x) can't be utilised.

Is there a method available that doesn't rely on some kind of matrix multiplication like the methods in the Broyden family?

 

[Strum]

I am not really sure I understand your question but how about a predictor corrector method?