A treecode algorithm based on tricubic interpolation

Treecode algorithms efficiently approximate N-body interactions in $O\left(N\right)$ or $O\left(N\text{log}N\right)$. In order to treat general 3D kernels, recent developments employ polynomial interpolation to approximate the kernels. The polynomials are a tensor product of 1-dimensional polynomials. Here, we develop an $O\left(N\text{log}N\right)$ tricubic interpolation based treecode method for 3D kernels. The tricubic interpolation is inherently three-dimensional and as such does not employ a tensor product. The form allows for easy evaluation of the derivatives of the kernel, required in dynamical simulations, which is not the case for the tensor product approach. We develop both a particle-cluster and cluster-particle variants and present results for the Coulomb, screened Coulomb and the real space Ewald kernels. We also present results of an MD simulation of a Lennard-Jones liquid using the tricubic treecode.