From SPME to P3M and back: a unified view on the accuracy and efficiency of particle-mesh methods

In computer simulations of bio-systems, charged soft matter, plasmas, and many more areas, advanced particle-mesh methods such as ParticleParticle Particle-Mesh (P3M) [1], Particle Mesh Ewald (PME) and Smooth Particle Mesh Ewald (SPME) [2], are commonly used to compute efficiently the long-range electrostatic interactions under periodic boundary conditions. These methods reduce the complexity of computing O(N) pairwise interactions into an O(N logN) problem by discretizing the charge density onto a mesh and by using Fast-Fourier-Transforms to determine the electrostatic potential. The various particle-mesh methods are intimately linked: they vary mainly in the way the charge density is assigned onto the mesh and the way forces are interpolated from the mesh. All particle-mesh methods can be understood within the same mathematical framework [1, 3]. Using this fact, we show how the accuracy of the SPME method, and that of its interlaced variant (staggered SPME [4]) can be predicted as function of the method’s parameters. An analytical formula is furthermore given to substract the spurious self-forces that appear in methods that use the analytical differentiation (AD) scheme for computing forces, like SPME and P3M-AD.