### Time integration again

In the post time integration: incompressible flow I explained a scheme example using PFEM. I don’t think it’s really “meshless”, even though by definition of same group it is. Personally, I more tend to see it a FEM with time-updated elements and shape functions, at lease mathematically. Anyway, the nice way of time spitting is what interests me here. Within an iteration, the variables are found following:

$\cdots\longrightarrow u_i^n, p^n\longrightarrow u_i^{\ast}\longrightarrow p^{n+1}\longrightarrow u_i^{n+1}\longrightarrow\cdots$

For each step above, equation for the unknowns to be solved are: Elliptic equation for $u^{\ast}$, elliptic equation for $p^{n+1}$ and algebra/finite difference for $u_i^{n+1}$. This means for this particular time updating plan, a single linear elliptic equation solver can be used. Remember, this virtue is due to two things, the first is the MEL to eliminate the annoying nonlinear convection term, second, the time splitting using $u^{\ast}$ as the intermediate variable. In summary, this scheme, as many other ones, gains the sequential solution by paying the price with solving one more variable (implicitly). It can be seen that within one iteration the solving of field equation costs most of the computing, while the choice of time integration scheme is the main source of stability issues.