Numerical investigation of three-dimensional shock focusing effects by an invariant difference scheme

Symmetry preserving difference schemes which are found on the basis of the Lie-group theory for partial differential equations applied to their first differential approximations are used to simulate the evolution and reflection of shock waves in a cubical cavity with rigid walls and walls with outflow boundary conditions. The explicit difference scheme was used for a shock capturing technique to find and trace the discontinuities in the flow field. The numerical model is based on the strong conservative form of the three-dimensional time-dependent Euler equations with an equation of state for an ideal gas. The three-dimensional time-dependent shock waves which are created by a centered gas volume under high pressure interact with the plane walls of the cavity and lead to focusing effects after the explosion. To follow the propagation of strong plane or spherical shock waves the shock capturing method was proved to be sufficient concerning the spatial resolution of the discontinuities. When interacting with the rigid walls of the cavity the shock waves are assumed to be reflected specularly and to interact with shock waves and contact discontinuities moving from the center to the edges and corners of the cavity. The symmetry preserving character of the invariant difference scheme under use was proved numerically by calculations over long time. No significant deviation of the symmetry character of the imposed initial distribution was found. Additionally, a numerical analysis of the conservation of energy in the cube was performed to find the behavior of the energy in time.