Concluding Remarks

This paper presented a survey of the different approaches to simulating electrostatic point charge systems in periodic boundary conditions via Ewald sums. The paper has reviewed the popular approaches along with some recent state of the art methods that handle Ewald sums more efficiently, i.e. Fourier- and multipole-based methods. Fourier-based techniques perform the ``true'' Ewald sum and rely on the reformulation of the reciprocal-space sum into a form that is effectively calculated using FFT in method. Multipole-based methods, theoretically algorithms, have been suggested as feasible alternatives to the Ewald sum. The computational efficiency and accuracy of multipole-based methods place them as strong contenders against today's fastest true Ewald sum methods. Ewald summation methods that truly evaluate the infinite sum will remain a favorable approach in the MD community as a well established simulation technique. Furthermore, the Ewald sum is still considered more suitable for crystalline structures than any other method. Presently, there is no conclusive evidence as to which of the methods (Fourier-based Ewald or periodic FMA) have better performance as such comparisons are strongly dependent on the implementation of the algorithm and the optimizations for a particular computer. Reports of the break-even point, i.e. the the number of particles at which the two methods are equally fast, have ranged from N=300 [19], to N=30,000 [49]. Recently, a break-even point of N=100,000 between FMA and a direct implementation of Ewald summation was reported by [22]. To put Fourier- and Multipole-based Ewald methods in perspective, we have tabulated sample simulation results as reported in the literature, Table[1].

From this account and our experience in evaluating long-range electrostatics we offer the following concluding remarks:

• Algorithms from all three approaches (standard, Fourier, and Multipole-based Ewald sum) are being used in MD ``production'' packages. Standard Ewald sum implementations, section(3), are relatively the easier to program, while Multipole-based algorithms, section(5), are more difficult to program with Fourier-based methods, section(4), somewhere in the middle.
• Fourier- and Multipole-based methods are both competitive approaches for evaluating the Ewald sum. Small systems, particles, can be efficiently simulated with standard Ewald sum techniques. For system sizes on the order of , Fourier-based methods are likely to be faster, whereas for system sizes of and above, Multipole-based methods are probably more efficient.
• Particle systems that are inherently periodic, e.g. crystals, can be simulated highly accurately with Fourier-based methods very efficiently.
• The Fast Multipole Algorithms, have been extended to compute the Leonard-Jones potential [21] and Polarization [35] which are likely to be further extended for systems with PBC. This will provide Multipole-based methods with the flexibility to simulate more realistic potentials for periodic particle systems. It is worth mentioning that the PME method has been extended to include the Leonard-Jones potential [15] as well.