Fast methods for long-range interactions in many-particle simulation

Computer simulations of complex particle systems play an increasingly important role across a broad range of disciplines in the natural and engineering sciences: for example, in astrophysics, plasma physics, material sciences, physical chemistry, biophysics and fluid dynamics, to name a few. It turns out that some of the most of interesting and important physical phenomena found in such systems also involve electrostatic, gravitational or hydrodynamic effects, where the proper inclusion of long-range interactions is essential to describe their equilibrium properties or dynamics correctly. This quickly becomes very expensive, in principle requiring the summation over O(N^2) interaction pairs for open systems, or involving infinite lattice sums in periodic systems. To render such problems numerically feasible, a number of efficient mathematical algorithms have been developed over the past 3 decades which reduce this algorithmic complexity down to O(N log N) or even O(N). The aim of this course is to introduce these so-called ‘fast Coulomb methods’ – starting with the well-known Ewald summation for periodic lattices, then covering advanced techniques such as P3M and real-space multipole methods. Practical hands-on sessions will give students a chance to develop their own algorithms, compare the performance of different solvers on typical benchmark problems, and to try their hand at real particle simulations with the help of a recently published parallel library