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!


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