Posted by Simon Wang on April 22, 1998 at 23:25:37:
Explicit and unconditionally stable algorithms for PDEs (2)
First of all, am I rediscovering the wheel? Let us look at the
Finite Element method. I believe the FE method is the best for problems
in Solid and Structural Mechanics but not for PDEs, esp. for parabolic
and hyperbolic equations (I have not got a conclusion for elliptic
equations). Generally speaking, we can not give accurate and explicit
equations globally for the former. It seems that the FE method is the
only choice in solving mechanical problems.
It has been considered that the No. 1 problem in solving PDEs
is to approach irregular boundaries (In textbooks, this is frequently
used to illustrate the defect of the classical FD method). Based on
this understanding, the FE method is used everywhere. The questions
are: if the area is a regular one, say rectangular/cubic, is the FE
the most efficient method?
let's look at the major drawback of the FE method:
The method is based on a global approaching. Hence, to generate
and solve global simultaneous equations is inevitable. It is also due
to this global approaching, any 'local' variation would lead to 'global'
changes (the simultaneous equations). Therefore, we have to deal with
large matrices; we need huge memory and fast processors. This is
particularly true for time-dependent problems. Hence, I will focus on
diffusion equations as there is a discontinuity problem for hyperbolic
equations for which I developed an adaptive algorithm by making use of
its unconditional stability.
Now, return to the original problem--irregular areas. A well
known fact that has been ignored for many years (or not drawn enough
attention) is that we have irregular-area problems simply because we
express the problem in Cartesian coordinate systems (physical domains).
The shape of the area changes with the use of different coordinate
systems. Therefore, almost all of the problems involving PDEs can be
considered to be in regular areas if coordinate transformations are
used. This is just the idea of the Numerical Grid Generation method.
However, to solve a PDE involving time variable in a regular area,
the classical implicit difference method is not the best one as
simultaneous equations are still required to generate and solve. I
believe this is one of the main reasons why the Numerical Grid
Generation method is not as popular as the FE method (How about
explicit and unconditionally stable FD method?).
I have developed some new algorithms to solve the 'transformed'
equations as stated in my previous message. Numerical test shows that
these new algorithms are, in addition to saving on required memory, at
least tens of times faster than the FE method in solving time-dependent,
such as the diffusion problems.
Maybe I need to say sth about the problem in the spectral
stability analysis method (my research topic 4): 'GLOBAL' analysis
(Fourier expansion of numerical solution) results in a LOCAL
amplification factor. This conflict can be seen more clearly when
considering PDEs with variable-coefficients. If interested, please
email me.
The conclusion about stability and applicability of the ADI
method is well known (Douglas, 1964). The method is considered suitable
only to problems in rectangular/cubic areas due to stability problem.
However, I developed an ADI method to solve the diffusion equations in
polar/cylindrical coordinate systems in 1991. The developed method
possesses unconditional stability. It is this that leads to my present
research. My later study revealed why this conclusion is incorrect.
This is just because of what I mentioned in the beginning: the shape of
the area can always be regarded as regular; influence of irregular
boundaries can be studied by the relative sizes of mixed and first
derivatives in the general expressions of PDEs.
The importance of this research can be readily seen. I have been
doing research on this project for several years by myself alone. I hope
this research can be accelerated and the new methods can be applied to
as many as possible practical problems. This is one of the main reasons
why I am looking for people who may be interested in this research. The
email address given last time is going to be cancelled. Please use:
XIN.WANG@QUB.AC.UK
or
MEG1839@A1.QUB.AC.UK
if you want to contact with me. Discussion and query on this research
is welcome.
Simon