A parallel integration method for solar system dynamics

We describe how long-term solar system orbit integration could be implemented on a parallel computer. The interesting feature of our algorithm is that each processor is assigned not to a planet or a pair of planets but to a time-interval. Thus, the 1st week, 2nd week,..., 1000th week of an orbit are computed concurrently. The problem of matching the input to the (n+1)-st processor with the output of the n-th processor can be solved efficiently by an iterative procedure. Our work is related to the so-called waveform relaxation methods in the computational mathematics literature, but is specialized to the Hamiltonian and nearly integrable nature of solar system orbits. Simulations on serial machines suggest that, for the reasonable accuracy requirement of 1" per century, our preliminary parallel algorithm running on a 1000-processor machine would be about 50 times faster than the fastest available serial algorithm, and we have suggestions for further improvements in speed.Comment: Submitted to AJ, 12 page

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Improved Hamiltonian for Minkowski Yang-Mills Theory

I develop an improved Hamiltonian for classical, Minkowski Yang-Mills theory, which evolves infrared fields with corrections from lattice spacing $a$ beginning at $O(a^4)$. I use it to investigate the response of Chern-Simons number to a chemical potential, and to compute the maximal Lyapunov exponent. Both quantities have small $a$ limits, in both cases within $10\%$ of the limit found using the unimproved (Kogut Susskind) Hamiltonian. For the maximal Lyapunov exponent the limits differ by about $5 \%$, significant at about $5 \sigma$, indicating that while a small $a$ limit exists, its value is corrupted by lattice artefacts. For the response of Chern-Simons number the statistics are not good enough to resolve $5 \%$ differences, but it seems possible in analogy with the Lyapunov exponent that the final answer depends on the lattice regulation.Comment: Latex, 33 pages plus 2 .epsi figures included with psfig. Revised to include new data which weakens some original conclusion