Employing GPUs for numerical integration with Lie-series

András Pál (Konkoly Observatory)


Abstract


The method of Lie-integration is a very effective algorithm for numerical solution of ordinary differential equations. The principle of the algorithm is to compute the coefficients of the Taylor-series 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 right-hand side of the differential equation), has better error propagation properties and the ``side-effect'' of knowing the analytic expansion of the solution also allows us other kind of studies. The previously mentioned recurrence relations are known for the N-body 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 Lie-integration algorithm on GPU architectures. We briefly summarize other advantages of this numerical method that makes particularly suitable on GPU systems. For instance, how the fact that the computation of the recurrence relations (in the case of the N-body 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.

Paper ID: P108

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