- *Yes, it's a good step. But it implies that you get a restriction on
the time step size. *
Does that become severe even if I use the present time-step for all other
terms? In the \nabla ( D(u) u) term, I will replace only D(u) with the
previous time-steps and retain the u^{n} to be at the current time-step.
So, the time-step restriction should not be as bad as the fully explicit
case, right?
- *You can also replace D(u^n) by D(u^*) where u^* is extrapolated from
the previous two or three time steps. That's a more accurate approximation
than just using u^{n-1}. *
That sounds perfect. Thank you. Is there any suitable functions from
deal.ii that I can use for this extrapolation? Is there any tutorial that
already does something this?
Regards,
Krishna
On Friday, March 13, 2020 at 7:34:51 PM UTC, Wolfgang Bangerth wrote:
>
> On 3/13/20 12:59 PM, Krishnakumar Gopalakrishnan wrote:
> > I have a non-linear diffusion equation of the form
> >
> > du/dt = \nabla.( D(u) \grad u)
> >
> > The non-linearity appears because of the dependence of the diffusion
> > coefficient on the solution.
> >
> > When discretising by the Rothe method, applying backward Euler method in
> > the strictest sense:
> >
> > (u^n - u^{n-1})/k^n - \nabla. ( D(u^n) ) * \grad{u^n}) = 0
> >
> > This would require Newton iterations and such complications. Is it okay
> > to simply use the numerical value of D from the previous time-step?
> >
> > (u^n - u^{n-1})/k^n - \nabla. ( D(u^{n-1}) ) * \grad{u^n}) = 0
> >
> > In this case, we get a nice linear equation, and most of step-26 can be
> > used as. D(u) is a continuous function of u. Is this semi-explicit type
> > of usage of diffusion coefficient, a reasonable way to tackle this
> problem?
>
> Yes, it's a good step. But it implies that you get a restriction on the
> time step size.
>
> You can also replace D(u^n) by D(u^*) where u^* is extrapolated from the
> previous two or three time steps. That's a more accurate approximation
> than just using u^{n-1}.
>
> Best
> W.
>
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
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