### Time integration in a dynamical FEM prob.

For any dynamical problems involving spatial evolution, each step is to be solved by proposed spatial discretization scheme. But unlike steady/static problem, which requires no particular time updating, dynamical problem requires discretization in one more dimension, the time.

This leads to one of the most important and, to me,  most messy part of numerical simulation. Say,  at any time $t_1$, spatial problem is (to skip the boundary condition)

$\mathcal{L}\phi(x,t_1)=0$

with time evolusion equation

$\frac{Du(x,t)}{Dt}=\mathcal{F}u(x,t)$

in which both $\mathcal{L}$ and $\mathcal{F}$ are global operators in the spatial domain. At each time like $t_1$, the first equation is to be solved, in discretized version, one or another. And the second equation is used to formulate the spatial domain in next step, after the local velocity $u$ is resolved and used to update the spatial configuration using

$x(t_2)=x(t_1)+u\Delta t$

Here comes the tricky part. In equation above, which time to use for velocity? In particular, should one use $u(x(t_1),t_1)$ or $u(x(t_2),t_2)$? And to the next step, when one tries to solve (this is so called “time integration”, because it’s discrete version of taking integral) differential equation of $u$, what time should be introduced on RHS?

There are various time updating plans in use, which can be catogorized into implicit and explicit families, depending on whether the unresolved variable at next time step show up on RHS. And the story continues in balancing the advantage/disadvatange of those catogories.