  In this paper we present a mathematical theory and a numerical method to
study the propagation of a three-dimensional (3-D) weak shock front into a
polytropic gas in a uniform state and at rest, though the method can be
extended to shocks moving into nonuniform flows. The theory is based on the use
of 3-D kinematical conservation laws (KCL), which govern the evolution of a
surface in general and a shock front in particular. The 3-D KCL, derived purely
on geometrical considerations, form an under-determined system of conservation
laws. In the present paper the 3-D KCL system is closed by using two
appropriately truncated transport equations from an infinite hierarchy of
compatibility conditions along shock rays. The resulting governing equations of
this KCL based 3-D shock ray theory, leads to a weakly hyperbolic system of
eight conservation laws with three divergence-free constraints. The
conservation laws are solved using a Godunov-type central finite volume scheme,
with a constrained transport technique to enforce the constraints. The results
of extensive numerical simulations reveal several physically realistic
geometrical features of shock fronts and the complex structures of kink lines
formed on them. A comparison of the results with those of a weakly nonlinear
wavefront shows that a weak shock front and a weakly nonlinear wavefront are
topologically same. The major important differences between the two are
highlighted in the contexts of corrugational stability and converging shock
fronts.
