Thanks for your explanations in details，that's exactly what I want.
Best
M.
Wolfgang Bangerth <bange...@colostate.edu> 于 2018年12月29日周六 上午12:33写道：
> On 12/28/18 1:14 AM, llf m wrote:
> > Yes, you are right. At first in Step-15
> > pseudo-timestepping(Lecture 31.7),
> > it linearized the nonlinear term that makes me confused, linearized a
> > nonlinear term
> > is a general approach? Is it acceptable? But finally I get the idea,
> yes, we need
> > linearization in nonlinear iteration.
> > In a nonlinear time-dependent problem F[u(t)]=0:
> > Time iteration:
> > Newton-Raphson iteration:
> > solve linearized equation
> F'[u(t_n)^k]delta_u =
> > - F(u(t_n)^k)
> > (while k is the index of newton
> > iteration,u(t_n)^0=u(t_{n-1}))
> > update u(t_n)^{k+1} = u(t_n)^{k} + delta_u
> > check convergence e.g.,
> norm(F(u(t_n)^k))<1e-10
> > if satisfied go to next time iteration
> > else go to next newton iteration
> > The above algorithm is the idea I get, maybe I still have some
> misunderstanding.
>
> That's exactly the idea. In each time step, you would want to start with
> something like
>
> u(t_n)^0 = u(t_{n-1})^*
>
> where the right hand side is the final Newton iterate of the previous time
> step. This gives you a reasonably good starting guess and you will not
> need a
> lot of Newton iterations in time step n. An even better starting guess
> would
> be to *extrapolate* from the previous time steps to the current time step,
> e.g., using
>
> u(t_n)^0 = u(t_{n-1})^* + [u(t_{n-1})^* - u(t_{n-2})^*]/dt * dt
> = 2*u(t_{n-1})^* - u(t_{n-2})^*
>
> where in the first line on the right, the term [...]/dt is an
> approximation of
> the time derivative, and consequently the right hand side of the first
> line is
> simply a linear extrapolation from the previous two time steps. Of course,
> if
> you wanted to go back in time, you can come up with even better starting
> guesses. With such a starting guess, you typically only need two or three
> Newton iterations per time step.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bange...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
> --
> The deal.II project is located at http://www.dealii.org/
> For mailing list/forum options, see
> https://groups.google.com/d/forum/dealii?hl=en
> ---
> You received this message because you are subscribed to the Google Groups
> "deal.II User Group" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to dealii+unsubscr...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.
>
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups
"deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to dealii+unsubscr...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.
