A Necessary Condition for Individual Time-Steps in SPH Simulations

TL;DR

This paper identifies a fundamental limitation of individual time-step schemes in SPH simulations for modeling explosions, and proposes a time-step limiter to ensure accurate shock handling.

Contribution

It introduces a necessary condition and a time-step limiter to improve the accuracy of SPH simulations involving strong explosions.

Findings

Without the limiter, SPH fails to handle shocks correctly.

The limiter ensures particles respond appropriately to shocks.

Improves simulation accuracy in multiphase interstellar medium models.

Abstract

We show that the smoothed particle hydrodynamics (SPH) method, used with individual time-steps in the way described in the literature, cannot handle strong explosion problems correctly. In the individual time-step scheme, particles determine their time-steps essentially from a local Courant condition. Thus they cannot respond to a strong shock, if the pre-shock timescale is too long compared to the shock timescale. This problem is not severe in SPH simulations of galaxy formation with a temperature cutoff in the cooling function at $10^4 {\rm K}$, while it is very dangerous for simulations in which the multiphase nature of the interstellar medium under $10^4 {\rm K}$ is taken into account. A solution for this problem is to introduce a time-step limiter which reduces the time-step of a particle if it is too long compared to the time-steps of its neighbor particles. Thus this kind of time-step constraint is essential for the correct treatment of explosions in high-resolution SPH simulations with individual time-steps.