How do we compute f(q_{k+1}) before we actually get q_{k+1}? I am a bit confused here.
ericchan
What is the reason for this being called "backward" Euler? I don't see anything conceptual about this method that has to relate to doing something backwards.
anonymous_panda
Maybe we know f(q_{k+1}) in advance?
nrauen
I think the idea is that in one case you take the current information and predict forward into the future and find what the resultant would be, where as backwards Euler you are essentially guessing at the new configuration as you also need it to take the velocity there and map it backwards to the original configuration.
keenan
@yongchi1 Terrific question---this is exactly what makes backward Euler harder to evaluate! The point is that you now have to solve this equation for $q_{k+1}$. For instance, suppose that $q$ is just a single scalar value, and $f(x)$ is just a super easy linear function $f(x) := ax+b$. Then your update is
$q_{k+1} = q_k + \tau (aq_{k+1} + b)$
You can then solve this equation directly for $q_{k+1}$ to get an update rule
$q_{k+1} = (q_k + \tau b)/(1 - a\tau).$
In general, however, it's not so easy: there may be no closed-form solution to the update equation. In this case, you have to use a numerical strategy like Newton's method to compute the update.
EdCat
What if we have multiple solution after solving the equation?
How do we compute f(q_{k+1}) before we actually get q_{k+1}? I am a bit confused here.
What is the reason for this being called "backward" Euler? I don't see anything conceptual about this method that has to relate to doing something backwards.
Maybe we know f(q_{k+1}) in advance?
I think the idea is that in one case you take the current information and predict forward into the future and find what the resultant would be, where as backwards Euler you are essentially guessing at the new configuration as you also need it to take the velocity there and map it backwards to the original configuration.
@yongchi1 Terrific question---this is exactly what makes backward Euler harder to evaluate! The point is that you now have to solve this equation for $q_{k+1}$. For instance, suppose that $q$ is just a single scalar value, and $f(x)$ is just a super easy linear function $f(x) := ax+b$. Then your update is
$q_{k+1} = q_k + \tau (aq_{k+1} + b)$
You can then solve this equation directly for $q_{k+1}$ to get an update rule
$q_{k+1} = (q_k + \tau b)/(1 - a\tau).$
In general, however, it's not so easy: there may be no closed-form solution to the update equation. In this case, you have to use a numerical strategy like Newton's method to compute the update.
What if we have multiple solution after solving the equation?