Numerical simulations of composite supernova remnants for small $\sigma$ pulsar wind nebulae

Composite supernova remnants consist of a pulsar wind nebula located inside a shell-type remnant. The presence of a shell has implications on the evolution of the nebula, although the converse is generally not true. The purpose of this paper is two-fold. The first aim is to determine the effect of the pulsar's initial luminosity and spin-down rate, the supernova ejecta mass, and density of the interstellar medium on the evolution of a spherically-symmetric, composite supernova remnant expanding into a homogeneous medium. The second aim is to investigate the evolution of the magnetic field in the pulsar wind nebula when the the composite remnant expands into a non-uniform interstellar medium. The Euler conservation equations for inviscid flow, together with the magnetohydrodynamic induction law in the kinematic limit, are solved numerically for a number of scenarios where the ratio of magnetic to particle energy is $\sigma < 0.01$. The simulations in the first part of the paper is solved in a one-dimensional configuration. In the second part of the paper, the effect of an inhomogeneous medium on the evolution is studied using a two-dimensional, axis-symmetric configuration.