For chaotic systems, one often uses variational equations to study the evolution of nearby trajectories.
Variational equations have better numerical properties as shadow particles and have been implemented in REBOUND for a while.
Dan Tamayo and I recently published a paper in which we describe how to generalize the conept of variational equations to second order.
We provide an implementation for the IAS15 integrator in REBOUND that lets you use second order variational equations very easily.
The possible applications are very exciting, ranging from better orbit fitting methods for explanets to spacecraft trajectory optimization methods.
Newton's Law of universal gravitation describes how any two objects in the universe attract each other.
Among many applications, it allows astronomers to calculate how planets, asteroids and comets move through the Solar System.
Although this law and the corresponding mathematical equations have been known for over three hundred years, how to calculate the solutions has been an active field of research for both astronomers and mathematicians alike ever since.
The equations are what mathematicians call transcendental, in other words, they have no simple solutions.
With the advent of computers, scientists were finally able to solve the equations at least approximately for a wide range of problems.
However, because the solutions are not exact, they cannot be used to predict the positions of planets very long into the future.
This is because the relatively small errors made on short timescales (say 1 year) quickly add up to large errors after long time scales (100 years). Very much like the weather report might give a fairly good prediction for the weather tomorrow, but not so for the weather a year from now. Astronomers are often interested in very long timescales, billions of years! So you see the problem, even a tiny error after 1 year, will have a dramatic effect after 1000000000 years.
Using various tricks from mathematics and computer-science, we were able to improve the standard method used for the last 30 years. It's a powerful class of algorithms that's called symplectic integrators. We made two significant improvements. First we were able to speed it up by a factor of 2 to 5.
Second, we were able to reduce the errors. We made sure that the errors don't add up as fast as they did in previous algorithms. Previously the error was growing by a factor of 100 by going from 1 year to 100 years. With our improvements, we can go from 1 year to 100 years and only improve the error by a factor of 10. We achieve a fundamental limit which says that what we have achieved is the most accurate algorithm possible. We cannot be any more accurate on a present day computer processors.
The paper describing WHFast is published in MNRAS and can also be found on the arXiv server.
The algorithm is part of the REBOUND package and can be downloaded at github.
My university has also written a press release about our results.
We also developed a new, high order integrator for gravitational dynamics that can handle both conservative and non-conservative forces.
We ran a lot of tests and all results seem to show that our algorithm is superior to other integrators, including mixed variable symplectic integrators such as the one by Wisdom and Holman and other high order non-symplectic integrators, for example Bulirsch-Stoer.
We took great care when implementing the scheme and achieve Brouwer's law (accuracy up to machine precision) for at least a billion orbits, equivalent to 10^11 timesteps. The scheme comes with adaptive timestepping and works extremely well on a wide range of problems with out-of-the-box setting and no fine-tuning of any parameters.
This includes long term integrations of planetary systems, close encounters, Kozai cycles, migrating planets and dust undergoing PR-drag. You can find the preprint at http://arxiv.org/abs/1409.4779.
Links to scientific publications
All of my publication are listed on this website. Alternatively, they can be found on the NASA Astrophysics Data System (ADS) or the arXiv preprint server.
I feel very strongly about open access.
All my publication are on the arXiv and you can access them free of charge.
If you also think publicly funded research should be freely available for everyone on this planet, please do the same.
Please also consider signing a petition against Elsevier.
Some of the CPS faculty, postdocs and students (I'm in the back on the left, Aleksandar Rachkov and Rejean Leblanc are next to me, Ari Silburt and Dan Tamayo are sitting at the table.
Current postdocs I'm closely working with:
Dan Tamayo (CITA/CPS).
Chin Chen (undergraduate research project),
Rejean Leblanc (working towards a MSc),
Ekin Ozturk (undergraduate research project),
Aleksandar Rachkov (working towards a PhD),
Vismay Shah (undergraduate research project),
Ari Silburt (working towards a PhD).