Comparison of treecodes for computing electrostatic potentials in charged particle systems with disjoint targets and sources

In molecular simulations, it is sometimes necessary to compute the electrostatic potential at M target sites due to a disjoint set of N charged source particles. Direct summation requires O(MN) operations, which is prohibitively expensive when M and N are large. Here, we consider two alternative tree-based methods that reduce the cost. The standard particle-cluster treecode partitions the N sources into an octree and applies a far-field approximation, whereas a recently developed cluster-particle treecode instead partitions the M targets into an octree and applies a near-field approximation. We compare the two treecodes with direct summation and document their accuracy, CPU run time, and memory usage. We find that the particle-cluster treecode is faster when N > M, that is, when the sources outnumber the targets, and conversely, the cluster-particle treecode is faster when M > N, that is, when the targets outnumber the sources. Hence, the two treecodes provide useful tools for computing electrostatic potentials in charged particle systems with disjoint targets and sources.