Multi-Level Ewald: A hybrid multigrid / Fast Fourier Transform approach to the electrostatic particle-mesh problem.

We present a new method for decomposing the one convolution required by standard Particle-Particle Particle-Mesh (P(3)M) electrostatic methods into a series of convolutions over slab-shaped subregions of the original simulation cell. Most of the convolutions derive data from separate regions of the cell and can thus be computed independently via FFTs, in some cases with a small amount of zero padding so that the results of these sub-problems may be reunited with minimal error. A single convolution over the entire cell is also performed, but using a much coarser mesh than the original problem would have required. This "Multi-Level Ewald" (MLE) method therefore requires moderately more FFT work plus the tasks of interpolating between different sizes of mesh and accumulating the results from neighboring sub-problems, but we show that the added expense can be less than 10% of the total simulation cost. We implement MLE as an approximation to the Smooth Particle Mesh Ewald (SPME) style of P(3)M, and identify a number of tunable parameters in MLE. With reasonable settings pertaining to the degree of overlap between the various sub-problems and the accuracy of interpolation between meshes, the errors obtained by MLE can be smaller than those obtained in molecular simulations with typical SPME settings. We compare simulations of a box of water molecules performed with MLE and SPME, and show that the energy conservation, structural, and dynamical properties of the system are more affected by the accuracy of the SPME calculation itself than by the additional MLE approximation. We anticipate that the MLE method's ability to break a single convolution into many independent sub-problems will be useful for extending the parallel scaling of molecular simulations.