Table 1 Symbols and definitions for the accuracy parameters of the \(\pmb{\mathrm{P}^{3}\mathrm{T}}\) scheme

α

timestep softening. For all runs α = 0.1

γ

ratio of inner and outer cutoff radius (\(r_{\mathrm{in}}/r_{\mathrm{cut}}\)). For all runs γ = 0.1

\(\Delta r_{\mathrm{buff}}\)

width of the buffer shell. \(\Delta r_{\mathrm{buff}}=3\sigma\Delta t_{\mathrm{soft}}\), as a standard value

\(\Delta t_{\mathrm{soft}}\)

timestep of the soft part. \(\Delta t_{\mathrm{soft}}=(1/256)(N/16\mathrm{K})^{-1/3}\), as a standard value

\(\Delta t_{\max}\)

maximum timestep of the hard part. \(\Delta t_{\max}=\Delta t_{\mathrm{soft}}/4\), as a standard value

ϵ

plummer softening length. ϵ = (4/N), as a standard value

η

accuracy parameter for timestep criterion. η = 0.1, as a standard value

\(r_{\mathrm{cut}}\)

outer cutoff radius of smooth transition functions W and K. \(r_{\mathrm{cut}}=4\Delta t_{\mathrm{soft}}\), as a standard value

\(r_{\mathrm{in}}\)

inner cutoff radius of smooth transition functions W and K (\(r_{\mathrm{in}}=\gamma r_{\mathrm{cut}}\))

θ

opening criterion for tree. θ = 0.4, as a standard value