Thursday, October 28, 2004

Hush, hush, don't rush

What is wrong with my long-term memory? I seem to have forgotten too many things from the past. Well, not forgotten, but they're hidden in some remote, dusty corner of my brain. Specifically, I mean mathematical knowledge here: What were properties of Crank-Nicholson, what's the use of a positive definite matrix, and why, oh why do the Navier-Stokes equations look only remotely familiar?

Have I been out of it for too long? Well, two years of computer science cannot be that devastating, can they?
Were the former Computational Science students educated in a too broad, but too little `in depth' fashion? Possibly, from database programming, along electrodynamics, to numerical PDE analysis, we've had it all. Still, most physics and mathematics courses were good enough. The fact that I'm able to realize that I've almost forgotten so many things indicates that I know of the existence of all these many things.
I think I know the true answer: I'm rushing through things too much. Get into any matter quickly, solve some problems for it, implement it, or write a short report on it, and proceed to what's next. Some computer science courses even included an 'open-book exam', which allowed me to pass without having studied the matter in advance at all. Common sense is a good thing to have, but to have more factual knowledge would also be very good.

So at the moment I'm taking a fresh-up course in computational fluid dynamics, and one in numerical linear algebra. For the latter, I don't have time now, caused by two courses in linear algebra and calculus that I teach, and one in scientific computing that I assist in. Although cfd is interesting, I feel the urge of doing my own research, some serious deadlines coming up within months and I've only just started. Or do we sense my old "rush-on-to-what's-next"-mistake again here? I guess so...

The problem is: when I see interesting things lying ahead of me, I get enthusiastic. At the moment I would love to start implementing and experimenting right away, whereas it is better to first catch up with the field by reading others' papers. That's what I'm doing right now. Fortunately I have some open ends in my own previous work, which I can work on even now already, gives some variety from day to day.

Analyzing oneself is not too difficult, but changing one's bad habits is a whole different story. I hope I'll succeed!

Friday, October 15, 2004

The Mesh Is Moving...

Maybe I should somehow 'mark' the beginning of a period of four years I just entered. October 1, 2004, I started as a PhD student at the Mathematical Department of Utrecht University in The Netherlands. Under supervision of Dr. Paul Zegeling, with Prof.dr. Henk van der Vorst as promoter, I work on a NWO-STW-funded project titled "Adaptive moving mesh methods for higher-dimensional nonlinear hyperbolic conservation laws".

Adaptive what??...

Adaptive meshes are space discretizations of a domain (here in either 1D, 2D or maybe later 3D), that are non-uniform. Instead of dividing the domain uniformly into N equally-sized cells, some cells are bigger than others. Parts of the solution that need a fine discretization will get small cells at that location, other locations will have bigger cells (coarser discretization) than a uniform discretization would yield.

But why bother?

Solving discretized systems can become very costly, the finer the discretization (the bigger N), the longer it takes. Instead of increasing N for obtaining a more accurate solution, let's keep it constant and shift each of the mesh points to an appropriate location such that the points will be close to each other where necessary for an accurate solution and further away at other parts of the domain.
We now have some additional cost for shifting the mesh points, and interpolating the solution each time the mesh changes, but we save the huge costs of increasing N.

And you need four years for this?

There are many things to this. Firstly, how can we have the mesh being moved automatically in an optimal way? Next, how do we efficiently, but accurately interpolate the solution on the newly moved mesh? Quite some research has already been done on this. Next, apply this in two-dimensional domains, how to prevent your mesh from becoming too irregular, or even worse, degenerate? The discretization of PDEs on a non-uniform mesh also becomes a whole lot less trivial in 2D and 3D.
What can be said on the convergence and stability of this method? Any formal error-estimates?

So now your plans are?

Re-reading a lot! Having been away from any mathematics for almost two years, quite some knowledge has sunk away into the very far corners of my brain. I sometimes feel like a bachelor student in mathematics... Besides that a conference comes up, I should recollect some ideas we had in the last two years and make a paper out of it, prepare for the visit of Dr. Tao Tang who is from The Hong Kong Baptist University and has many experience with moving mesh problems involving systems with conservation laws. Looking forward to that.

Alright, that was quite some writing, maybe not even too interesting as it was a very quick and general story, but think of it as the 'obligatory introduction story'. Stay tuned 'till next time!

Sunday, October 10, 2004

"And here we are..."

Saturday, 8PM, and in a whim I decided to make myself a blogger-account. Not because of a sudden urge to spill my thoughts on the internet, but rather because I found blogger looking really decent when reading a new blog post by Martin Bravenboer. And since I always like new things that look great, I thought I should go for it as well.

Besides that, Martin mentions that "many bright people are communicating their ideas and thoughts in their weblogs" and he is surprised about how "limited the number of researchers that are blogging" is. The limited number probably depends on what area of research one is looking at, but the note on 'bright people' really did appeal to me, so here I am.

What will I blog about? Purely professional matter? In my case that would be numerical mathematics, since I just started as a PhD student on time-dependent adaptive meshes for PDE systems.
Or could this blog involve personal matter as well? Maybe on which nice new skate routes I found, or on which beautiful new photographs I shot today?
We'll see what happens, as said I just started in a whim and actually do not have anything bright to share with you at the moment, dear reader. So, please "enjoy your meal and we'd love to see you back next time!"