

Paper: 
Employing GPUs for Numerical Integration with Lieseries 
Volume: 
461, Astronomical Data Analysis Software and Systems XXI 
Page: 
69 
Authors: 
Pál, A. 
Abstract: 
The method of Lieintegration is a very effective algorithm for
numerical solution of ordinary differential equations. The principle
of the algorithm is to compute the coefficients of the Taylorseries
for the solution involving recurrence relations. This approach also
yields more possibilities for various types of adaptive integration
since not only the integration stepsize but simultaneously the
polynomial order of the power series expansion can also be altered.
In addition, alternation of the stepsize does not yield a loss in the
(expensive) computing time. The “disadvantage” of the method is
because of the recurrence formulae: these set of equations depends on
the particular problem itself and therefore had to be derived in
advance of the actual implementation. However, the method is
definitely faster than the classic known explicit methods (that
do not depend on the righthand side of the differential equation),
has better error propagation properties and the “sideeffect” of
knowing the analytic expansion of the solution also allows us other
kind of studies. The previously mentioned recurrence relations are
known for the Nbody problem, thus the dynamical analysis of
planetary systems could be made very effective. In this
presentation we discuss the questions and possibilities
related to the implementation of the Lieintegration algorithm
on GPU architectures. We briefly summarize other advantages of
this numerical method that makes it particularly suitable on GPU
systems. For instance, how the fact that the computation of the
recurrence relations (in the case of the Nbody problem)
needs only evaluating additions, subtractions and multiplications
can be exploited on GPUs. Initial works show that studies related
to exploration of the phase space (thus as stability studies,
where the similar dynamical system is investigated in the case
of various initial conditions) can be achieved rather efficiently.
Such studies are in the focus of astronomical research in
the case of both the Solar System and extrasolar
planetary systems as well. 



