I am reading the 2016 SIGGRAPH notes on MPM methods and I am trying to understand the part about implicit Euler updates. In particular the notes write these 2 equations:
$$h(v^{n+1}) = Mv^{n+1} - \Delta t f(x^n + \Delta t v^{n+1}) - Mv^n = 0$$ $$v^{i+1} = v^i - \bigg(\frac{\partial h}{\partial v}(v^i) \bigg)^{-1}h(v^i)$$
The first being a conservation of momentum constraint, the second being the implicit Euler update.
I understand the general idea, but I don't get what the actual computation I am supposed to do is.
i.e given a grid state $G$ having terms for velocity, pressure, mass,... how do I actually compute the per grid cell velocities of the next stage? Which calculations do I actually do?