A New Family of Regularized Kernels for the Harmonic Oscillator

TL;DR

This paper introduces a new two-parameter family of regularized kernels for the harmonic oscillator, improving high-order time stepping accuracy in N-body simulations through Taylor expansion-based derivations and numerical validation.

Contribution

It presents a novel family of regularized kernels derived from Taylor expansions, with proven validity and error estimates, enhancing simulation accuracy.

Findings

High-order kernels improve simulation precision.

Error estimates guide kernel selection.

Numerical experiments demonstrate benefits.

Abstract

In this paper, a new two-parameter family of regularized kernels is introduced, suitable for applying high-order time stepping to N-body systems. These high-order kernels are derived by truncating a Taylor expansion of the non-regularized kernel about $(r^2+\epsilon^2)$, generating a sequence of increasingly more accurate kernels. This paper proves the validity of this two-parameter family of regularized kernels, constructs error estimates, and illustrates the benefits of using high-order kernels through numerical experiments.