Numerical Analysis: Spectrally accurate fast Ewald summation

The goal of this project is to develop spectrally accurate and fast Ewald summation methods – suitable for parallel computations – for various problems in molecular dynamics and fluid mechanics.  Ewald summation is the established method to calculate long range interactions in molecular simulations under periodic boundary conditions. This includes the evaluation of non-bond interactions that are taken into account in bimolecular simulations, as well as other applications where potentials behave as 1/r, e.g. the Stokes equations. In [1] we develop a spectrally accurate and fast, O(N log N), method to address this problem for the Stokes case. Error bounds provide practical parameter choices such that the desired accuracy is attained.  All major software packages for molecular simulation (e.g. Gromacs, Amber, NAMD) include fast Ewald methods, from the established family (such as the well known PME, SPME and P3M methods), for electrostatic calculations. Our results indicate that our method as adapted to this case compares favorably to these, especially regarding memory [2]. One consequence of the spectral accuracy properties of our method is that much smaller grids are needed. The global nature of the FFT, which is at the core of these methods, limits parallel scalability, and this strongly favors smaller grids. In collaboration with the molecular simulation community, we want to evaluate the computational efficiency that a parallel version of our method can achieve as compared to the established methods.   Systems with planar periodicity arise in surface chemistry and membrane simulations, and Ewald type methods for such systems are much less mature than for the triply periodic case. Also in this case, we have initiated the design of a spectrally accurate FFT based method, and foresee that it will be highly competitive as compared to existing methods. The extension of this method to planar periodicity of fundamental solutions for Stokes flow will also be very useful as it can be applied to accelerate microfluidic simulations based on boundary integral formulations.

[1] Lindbo, D. and Tornberg, A.-K., Spectrally accurate fast summation for periodic Stokes potentials. J. Comput. Phys., 229:8994 – 9010, 2010.

[2] Lindbo, D. and Tornberg, A.-K., Spectral accuracy in fast Ewald-based methods for particle simulations. Submitted.