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.

-- 
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Wolfgang Bangerth          email:                 bange...@colostate.edu
                            www: http://www.math.colostate.edu/~bangerth/

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