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

[Feng-Chao Wang]

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.  

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 
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20111228/4760f960/attachment-0001.htm>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: not available
Type: image/jpeg
Size: 52349 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20111228/4760f960/attachment-0001.jpeg>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: equation.jpg
Type: image/jpeg
Size: 24483 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20111228/4760f960/attachment-0001.jpg>