A Rigorous Comparison Of The Ewald Method And The Fast Multipole Method In 2 Dimensions

The most efficient and proper standard method for simulating charged or dipolar systems is the Ewald method, which asymptotically scales as N-3/2 where N is the number of charges. However, recently the ''fast multipole method'' (FMM) which scales linearly with N has been developed. The break-even of the two methods (that is, the value of N below which Ewald is faster and above which FMM is faster) is very sensitive to the way the methods are optimized and implemented and to the required simulation accuracy.In this paper we use theoretical estimates and simulation results for the accuracies to carefully compare the two methods with respect to speed. We have developed and implemented highly efficient algorithms for both methods for a serial computer (a SPARCstation ELC) as well as a parallel computer (a T800 transputer based MEIKO computer). Breakevens in the range between N = 10000 and N = 30000 were found for reasonable values of the average accuracies found in our simulations. Furthermore, we illustrate how huge but rare single charge pair errors in the FMM inflate the error for some of the charges.