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