Comparison between particle and fluid approximations to dust dynamics Joanna Dr¸ a˙ zkowska 1 , Micha l Hanasz 1 , Kacper Kowalik 1 Abstract: We present a new particle module of the magnetohydrodynamic (MHD) Piernik code. The original multi-fluid grid code based on the Relaxing Total Variation Diminishing (RTVD) scheme has been extended by addition of dust described within the particle approximation. The dust is now described as a system of interacting particles. The particles can interact with gas, which is described as a fluid. In this poster we introduce the scheme used to solve equations of motion for the particles and present the first results coming from the module. The results of test problems are also compared with the results coming from fluid simulations made with Piernik-MHD code. The comparison shows the most important differences between fluid and particle approximations used to describe dynamical evolution of dust under astrophysical conditions. Keywords: hydrodynamics — methods: numerical — dust 1. Piernik-MHD Piernik is a multi-fluid grid MHD code based on the RTVD conservative scheme by Jin & Xin (1995) and Trac & Pen (2003). Piernik can be used to examine dynamics of ion- ized or neutral gas, as well as dust treated as a pressureless fluid. The code computes conservative fluid variables (fluid density, momentum, total energy density) for each cell of the grid. The basic scheme has been extended by addition of many facilities which are useful in astrophysical fluid- dynamical simulations, e.g. shearing-box boundary con- ditions, Ohmic resistivity module and selfgravity module. See Hanasz et al. (2008a,b,c, 2009) for more details. 2. Particle module Dust can be described in fluid and particle approximations. In the particle module of PIERNIK-MHD code the dust component is described as a system of independent parti- cles that can interact with each other. The particles can also interact with gas considered as a fluid. For each particle, equation of motion is solved using the scheme described in the next subsection. Scheme To solve the equation of motion for dust particles we use the scheme known as Verlet leap-frog method. In this algo- rithm, the velocities are calculated at time t + 1 2 dt and used to calculate the positions, r, at time t + dt. In this way, the velocities leap over the positions, then the positions leap over the velocities. Generally, the scheme can be noted as: r(t + dt)= r(t)+ v(t + 1 2 dt)dt, (1) v(t + 1 2 dt)= v(t - 1 2 dt)+ a(t)dt. (2) 3. Results To compare fluid and particle approximations applied for the dust component we carried out several test problem simulations with the same initial conditions applied in both approaches. 1 Toru ´ n Centre for Astronomy, Nicolaus Copernicus University, Toru ´ n, Poland The first test problem relies on the analysis of 1D sinu- soidal velocity perturbation. The fluid approximation re- sult, which can be veryfied by an analytical solution of the Burger’s equation (Toro, 1999), displays a conversion of the initial sinusoidal velocity profile into the sawtooth profile and then smoothing until the flat profile (figure 1). The dis- continuity in the velocity profile can be interpreted as shock front. Fig.1–Sinusoidal wave simulation result coming from the fluid approximation In the case of noninteracting particles the particle model leads to multiple velocity values in the velocity profile (fig- ure 2). To avoid the unphysical evolution of the particle sys- tem we have introduced interaction between particles. The interaction is analogous to inelastic collisions. The particles stick when they meet each other in the same grid cell. Fig.2–Sinusoidal wave simulation result coming from the particle model in case of noninteracting particles The result coming from the particle approach with inter- actions taken into account (figure 3) appears different than the result given by the fluid approximation, because the particles group together into clusters. In the fluid simula- tions all the physical quantities are computed for every cell of the domain, even if density is very small. In the particle simulations the values of physical quantities are specified only in the particles locations. The fluid density profile at the end of the fluid simulation is represented by one peak of density. Respectively, at the end of the particle simulation all the particles are grouped together into one aggregate. ICYA2009 proceedings 1 arXiv:1009.0627v1 [astro-ph.EP] 3 Sep 2010