Go to Infinity in Real: Lattice Sum With Multipole Expansions

Periodic boundary conditions are widely imposed with Poisson solvers for cosmological simulations. Due to the well-known conditional convergence of the 1/r potential, Ewald summation or its variants are often combined with tree codes by current N-body practitioners. However, with FFT to accelerate Fourier sum, aliasing due to the finite and discrete sampling produces force errors difficult to control. We revisit the lattice sum with multipole expansion, clarify how Ewald sum treats the conditional convergence problem,  and derive correction terms which are well suited for real-space solvers, such as BH tree or fast multipole method. We are able to impose periodic boundary conditions with extremely high accuracy without using any Fourier techniques. We present a run with 2048^3 particles in a 500 Mpc box using a new fast multipole code based on this approach.