Some question about explicit and implicit time ODE solvers

[Wayne901]

Some question about explicit and implicit time ODE solvers

[Wayne901]

I also wonder why there's no FormSystemMatrix in ex9 but is in ex16. Is it because the DG FE space?

[Wayne901]

Now I'm clear about the ImplicitSolve(), it's about resolving
$\frac{du}{dt} = -M^{-1}K(u+h\frac{du}{dt})$
=> $\frac{du}{dt} = -(M+hK)^{-1}Ku$
and in ode.cpp BackwardEuler updates $u_{n+1}$ as $u_{n+1} = u_{n} + h\frac{du}{dt} = u_n - h(M+hK)^{-1}Ku_{n}$.
This is different to the Backward Euler time discretization I mentioned in the first question, where I don't solve $\frac{du}{dt}$ first but solve $u_{n+1}$ directly as $u_{n+1} = (M+hK)^{-1}Mu_{n}$.
Then I have a question, which scheme should be applied in terms of less computational effort? Though I think the update $\frac{du}{dt}$ one should be used for convinience, since MFEM's ODESolver classes are designed based on it.