A non-periodic particle mesh Ewald method for radially symmetric kernels in free space

I will present the non-periodic particle-mesh Ewald (NPME) method for evaluating pair potentials of charges that interact through a radially symmetric kernel f(r) in free space. The calculation starts by splitting f(r) into a short-range part f_s(r) and a smooth long-range part f_l(r). The smooth contribution is represented with a Fourier extension obtained numerically by discrete-Fourier-transform interpolation; this numerical route removes the usual restrictions on kernel form, kernel splitting, and box shape, so the same machinery applies to the 1/r,r^α,exp(ik_0r)/r kernels in rectangular volumes. A derivative-matched (DM) kernel split is introduced for arbitrary radial kernels. For Coulomb interactions, DM splitting makes the direct sum roughly twice as fast and reduces the required FFT grid by a factor of two to three compared with the conventional Ewald split. An independent grid-storage scheme reduces the remaining grid memory by a further factor of four. Performance tests were run on a 32-core (64-thread) workstation. For Coulomb systems containing 10^4–10^7 charges, npme computes potentials about three times faster than a vectorized fmm3D implementation; for Helmholtz systems with 3×10^5–3×10^7 surface charges, the speed-up is roughly a factor of ten. (As an upper-size stress test, the code also completed a one-billion-charge water droplet without exhausting memory, though that case is not aimed at production molecular dynamics.) I will further outline a hybrid Method-of-Moments + NPME application that reproduces the radar cross section of a metal sphere with sixty million surface-current unknowns. The final part of the talk shows initial results for applying DM splitting to periodic PME–yielding similar speed-ups and grid savings–and sketches possible next steps, including a GPU implementation and a two-level PME scheme for large parallel jobs. Code, documentation, and examples are on GitHub; I welcome questions and technical discussions after the talk.