# Introduction

## Background

The interstellar medium (ISM)is a site for various astrophysical processes of interest, such as.

## Hydrodynamic Theory

### Shocks

We list several quantities that we will refer to frequently in this work \begin{aligned} {\cal M} &= \frac{u}{c_{\rm s}} \ , \\ \beta &= 0\end{aligned} The Rankine-Hugoniot relations are

\begin{aligned} \nabla{\cdot}(\rho {\bf u}) &= 0 \ , \\ \nabla{\cdot}(\rho{\bf u}{\bf u} + p) &= 0 \ , \\ \nabla{\cdot}[{\bf u}(e + p)] &= 0 \ .\end{aligned}

### Hydrodynamics instabilities

A few relevant hydrodynamic instabilities in shock-cloud interactions include Rayleigh-Taylor (RT), Richtmeyer-Morton (RM), and Kelvin-Helmholtz (KH) instabilities.

## Simulations

The physical implications of shock-cloud interactions have motivated numerical simulations to characterize the dynamics and evolution of this process. Fragile et al. (2005) have shown that radiative cooling can affect the dynamics of shock-cloud interactions described by MHD.

## Experiments

Experiments of shock-cloud interactions have been performed.

# Methods

In this section, we describe the code and methods that we use for this work. More detailed descriptions of the code can be found in the code manual.

## Code

We use the FLASH code (Fryxell et al. 2000) a massively parallel Eulerian hydrodynamical code with adaptive mesh refinement (AMR). The MHD solver is highly efficient and explicitly maintains that the magnetic field has zero divergence. This solver is based on a directionally unsplit method (Lee 2006) that uses staggered-mes differencing and constrained transport with multi-dimensional fluxes obtained from the solution of the MHD Riemann problem. The method also employs efficient dissipation controls to prevent the growth of instabilities that arise in traditional constrained-transport schemes. The solver also resolves shocks in 1-2 mesh zones, which is important in the context of CR propagation. FLASH also features a N-body particle solver for collision-less species, which will be useful for PIC simulations explored in future work. FLASH implements block-structure AMR using an oct-tree data structure that permits efficient parallel load balancing (MacNeice et al. 2000). To capture flow discontinuities, the code refine blocks based on the second derivative of the gas density and pressure.

## Governing equations

Our model is represented by the two-fluid MHD equations

\begin{aligned} \frac{\partial \rho}{\partial t} &+ \nabla{\cdot}(\rho {\bf v}) = 0 \ , \\ \frac{\partial (\rho {\bf v})}{\partial t} &+ \nabla{\cdot} \left(\rho {\bf v}{\bf v} - \frac{{\bf B}{\bf B}}{4\pi}\right) + \nabla p_{\rm tot} = 0 \ , \\ \frac{\partial e}{\partial t} &+ \nabla{\cdot} \left[(e + p_{\rm tot}){\bf v} - \frac{{\bf B}({\bf B}{\cdot}{\bf v})}{4\pi}\right] = 0 \ , \\ \frac{\partial{\bf B}}{\partial t} &- \nabla{\times}({\bf v}{\times}{\bf B}) = 0 \ ,\end{aligned}

where $$\rho$$ is the gas density, $${\bf v}$$ is the gas velocity, $${\bf B}$$ is the magnetic field, $$e$$ is the total energy density, which includes contributions from thermal, kinetic, and magnetic energy densities, $e = e_{\rm th} + \frac{1}{2} \rho v^2 + \frac{1}{8\pi} B^2 \ ,$ $$p_{\rm tot}$$ is the total pressure, which includes contributions from the gas and magnetic pressures $p_{\rm tot} = p_{\rm th} + \frac{1}{8\pi} B^2 \ .$ The equation of state for the thermal gas and is taken to be $p_{\rm th} = (\gamma_{\rm th} - 1) e_{\rm th} \ .$

## Hydrodynamic solvers

The FLASH code has two operator-splitting methods: directionally split and unsplit. The directionally split piecewise-parabolic method (PPM) uses second-order Strang time splitting and the new directionally unsplit solver is based on Monotone Upstream-centered Scheme for Conservation Laws (MUSCL) Hancock type second-order scheme.

### The piece-wise parabolic method

The PPM algorithm is described in detail in Woodward and Colella 1984 and Colella and Woodward 1984. The method is a higher-order version of the Godunov method (Godunov 1959). Godunov’s method uses a finite-volume spatial discretization of the Euler equations together with an explicit time discretization.

## The eight-wave solver

The eight-wave MHD solver is based on a finite-volume, cell-centered method that was proposed by Powell et al. 1999. This unit uses directional splitting to evolve the MHD equations and uses the truncation-error method, which effectively removes the effects of unphysical magnetic monopoles if they are generated during simulations. It does not, however, completely eliminate monopoles that are spurious in a strict physical law.