See this thread for a similar question with some responses:
http://www.csharphelp.com/board2/read.html?f=1&i=28559&t=28559

But if you have the money you could see if there are applications which
already do what you want.  For example flexpde is one i found:
http://www.pdesolutions.com/
Another option might be to use matlab, which you may or may not want to
interface from C#:
http://www.mathworks.com/support/solutions/data/1-X1PFC.html

But if you want to do all the coding yourself, here is some info if it
is of help.
If I'm understanding, you want to do use partial differential
equations, with graphical output, and take advantage of multiple
processors.

First, you might look at existing libraries that can handle partial
differential equations (PDE's), for example PETSc or EXPDE or others,
and see if they meet your requirements:
http://www.oonumerics.org/oon/
http://wotug.ukc.ac.uk/parallel/nhse/hpc-netlib/sw_eval/pdes/
http://www-unix.mcs.anl.gov/petsc/petsc-as/
http://www10.informatik.uni-erlangen.de/~pflaum/expde/public_html/index.html

Most of them are written in C or C++, but you can still use them from
C# and .NET, using the [DllImport] attribute.  You would need to create
a C#/.NET wrapper on top of whichever library you use.  Sort of like in
the matlab sample at the mathworks link I posted above.

For the multi-processor part, you might look into MPI:
http://www-unix.mcs.anl.gov/mpi/
PETSc (mentioned above) uses MPI, for example.
Here is a .NET wrapper for MPICH, an implementation of MPI:
http://www.osl.iu.edu/research/mpi.net/
http://www-unix.mcs.anl.gov/mpi/mpich/
Another implementation of MPI is lam-mpi:
http://www.lam-mpi.org/

And for the graphical output, you'll want to learn windows.forms.
There are some free graph plotting controls already out there like
NPlot and ZedGraph.