I have a question about something from my lecture:
$"$The mono-implicit Runge-Kutta-Method is given as $$k_i(u_j,u_{j+1})=f((1-\delta_i)u_j+\delta_iu_{j+1}+h\sum_{v=1}^{i-1}a_{iv}k_v(u_j,u_{j+1}))$$ $$u_{j+1}=u_j+h\sum_{i=1}^sb_ik_i(u_j,u_{j+1})$$ with $\delta_i, a_{iv}, b_i \in \mathbb{R}$ and $i=1,...,s$
This equals the regular definition of a Runge-Kutta-Method: $$k_i=f(t+c_ih, u+h\sum_{j=1}^{i-1}a_{ij}k_j)$$ $$u_{j+1}=u_j+h\sum_{i=1}^sb_ik_i$$ with $b_i, c_i, a_{ij} \in \mathbb{R}"$
My question: How do the methods above equal each other? I've tried to write it down, but I don't see them being similar. Also, something I've been wondering: What's the point of writing it that way? I don't see any advantages coming from writing it like that.
I appreciate any help.