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In this paper we present a mathematical theory and a numerical method tostudy the propagation of a three-dimensional (3-D) weak shock front into apolytropic gas in a uniform state and at rest, though the method can beextended to shocks moving into nonuniform flows. The theory is based on the useof 3-D kinematical conservation laws (KCL), which govern the evolution of asurface in general and a shock front in particular. The 3-D KCL, derived purelyon geometrical considerations, form an under-determined system of conservationlaws. In the present paper the 3-D KCL system is closed by using twoappropriately truncated transport equations from an infinite hierarchy ofcompatibility conditions along shock rays. The resulting governing equations ofthis KCL based 3-D shock ray theory, leads to a weakly hyperbolic system ofeight conservation laws with three divergence-free constraints. Theconservation laws are solved using a Godunov-type central finite volume scheme,with a constrained transport technique to enforce the constraints. The resultsof extensive numerical simulations reveal several physically realisticgeometrical features of shock fronts and the complex structures of kink linesformed on them. A comparison of the results with those of a weakly nonlinearwavefront shows that a weak shock front and a weakly nonlinear wavefront aretopologically same. The major important differences between the two arehighlighted in the contexts of corrugational stability and converging shockfronts.

Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws