Galerkin methods for odes

Question

Could you give me some information about the multi-adaptive Galerkin methods for odes??

What does the term "multi-adaptive" mean??

Are there real-world problems at which we could apply these methods??

Answer 1

Regarding all three of your queries, here is a very useful link that can answer them.

An Article Regarding Multi Adaptive Galerkin Methods For Lodes:

http://arxiv.org/pdf/1205.2750v1.pdf

In short:

Could you give me some information about the multi-adaptive Galerkin methods for odes?

See: http://arxiv.org/pdf/1205.2750v1.pdf / http://www.math.chalmers.se/~logg/pub/thesis/compact-disc/papers/pdf/paper-02.pdf

What does the term "multi-adaptive" mean?

Multi-Adaptivity: "Taking adaptivity one step further, to allow for individual time- steps, order and quadrature, so that in particular each individual component has its own time-step sequence."

Are there any real world problems at which we could apply these methods?

Yes of course.

"The ODE models a very large class of problems, covering many areas of applications. Often different solution components have different time-scales and thus ask for individual time-steps. A prime example to be studied in detail below is our own solar system, where the moon orbits around Earth once every month, whereas the period of Pluto is $250$ years. In numerical simulations of the solar system, the time-steps needed to track the orbit of the moon accurately are thus much less than those required for Pluto, the difference in time-scales being roughly a factor $3000$.

Surprisingly, individual time-stepping for ODEs has received little attention in the large literature on numerical methods for ODEs. For specific applications, such as the $n$-body problem, methods with individual time-stepping have been used —but a general methodology has been lacking. Our aim is to fill this gap. Flaherty has constructed a method based on the discontinuous Galerkin method combined with local forward Euler time-stepping. A similar approach is taken, where a method based on the original work by Osher and Sanders is presented for conservation laws in one and two space dimensions. Typically the time-steps used are based on local CFL conditions rather than error estimates for the global error and the methods are low order in time (meaning $≤ 2$). We believe that work on multi-adaptive Galerkin methods (including error estimation and arbitrary order methods) presents a general methodology to individual time-stepping, which will result in efficient integrators also for time-dependent PDEs."

ALSO

"We have applied mcG(q) and mdG(q) to a variety of prob- lems to illustrate their potential. In these applications, including the Lorenz system, the solar system, and a number of time-dependent PDE problems, we demonstrate the use of individual time-steps, and for each system we solve the dual problem to collect extensive information about the problems stability features, which can be used for global error control."

Hope this helps. :)