[petsc-users] Strongly nonlinear equation solved within the framework of PETSc

[Jed Brown]

On Tue, Dec 27, 2011 at 19:58, Feng-Chao Wang <wolfshow at gmail.com> wrote:

> **
> Dear all,
>
> I want to numerically solve a strongly nonlinear fourth-order equation
> (2-D), which is used to describe the dynamics of a liquid film.
> please find the form of the equation below or in the attachment, The
> thickness of the film,  H(X,Y,T) is the function to be solved, C, G, A0 are
> constant parameters.
>
>
> I wrote a PETSc programs for this problem, using CN method in time.
> However, it does not work well. In some cases, there are some
> negativevalues in the solution (Obviously, the thickness of the liquid film
> could not be negative.) While in some other cases, the solution remains as
> the initial condition, looking like the governing equation doesn't work at
> all.
>

1. Make sure that the equations are actually being solved, for instance
with -snes_monitor -snes_converged_reason -ksp_converged_reason.

2. Start with a time integration method that has a discrete maximum
principle. Implicit Euler has this property for many classes of systems,
but CN does not.

3. Consider writing the system as a variational inequality if positivity is
still a problem with the time steps you want to take. Then use SNESVIRS (or
SNESVIRSAUG or SNESVISS).


>
> I think it is because that the solution of this equation may form infinite
> fradient when the equation is developed.
>
> Some literatures reported that the similar problem was solved successfully
> using the ADI (Alternating Direction Implicit) method. Unfortunately, I
> found that the current PETSc doesn't support ADI.
>
> I also read a paper in which the similar equation was solved by using the
> CN scheme in time. Besides, the author employed the modified second-order
> upwind difference method to handle the nonlinear terms due to the inability
> of the centered differences in space.
>
> I prefer to use PETSc because this powerful toolkit enables easy parallel
> computation of PDE since I am not familar with MPI. Could anyone please
> give me some suggestions on how I can solve the above equation within the
> framework of PETSc?
>
> Thanks very much in advance!
>
> Feng-Chao Wang
>
> 2011-12-28
> ------------------------------
>
>
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