  Shock wave refraction at a sharp density interface is a classical problem in
hydrodynamics. Presently, we investigate the strongly planar refraction of a
magnetohydrodynamic (MHD) shock wave at an inclined density interface. A
magnetic field is applied that is initially oriented either perpendicular and
parallel to the motion of incident shock. We explore flow structure by varying
the magnitude of the magnetic field governed by the non-dimensional parameter
$\beta \in (0.5, 10^6)$ and the inclination angle of density interface $\alpha
\in (0.30, 1.52)$. The regular MHD shock refraction process results in a pair
of outer fast shocks (reflected and transmitted) and a set of inner nonlinear
magneto-sonic waves. By varying magnetic field (strength and direction) and
inclination interface angle, the latter waves can be slow shocks, slow
expansion fans, intermediate shocks or slow-mode compound waves. For a chosen
incident shock strength, and density ratio, the MHD shock refraction
transitions from regular (all nonlinear waves meeting at a single point) into
irregular when the inclined density interface angle is less than a critical
value. Since the MHD shock refraction is self-similar, we further explore by
converting the initial value problem (IVP) into a boundary value problem (BVP)
by a self-similar coordinate transformation. The self-similar solution to the
BVP is numerically solved using an iterative method and implemented using the
p4est adaptive mesh framework. The simulation shows that a Mach stem occurs in
irregular MHD shock refraction, the flow structure can be an MHD equivalent to
a single Mach reflexion irregular refraction $MRR$ and convex-forwards
irregular refraction $CFR$ that occur in the hydrodynamic case.
