Transmission of a Slowly Moving Shock across a Nonconservative Interface

Computational fluid dynamics using composite overlapping grids plays an important role in todayʹs fluid mechanics with complex flows. The key point in the overlapping grid method is how to ensure conservation for shock waves. This was first studied by M. Berger under the framework of weak solutions for vanishing mesh size, leading to the well-known flux interpolation interface condition (SIAM J. Numer. Anal.24, 967 (1987)). The present author used the Rankine–Hugoniot relation to directly analyze the transmission of a shock across the interface and showed that, for the scalar Burgers equation, a nonconservative treatment leads to correct transmission of shocks even for finite mesh sizes if the interior difference scheme contains enough dissipation, and that shock penetration trouble only occurs for very slowly moving shock waves (SIAM J. Sci. Comput.20, 1850 (1999)). This is reconsidered here for the system of Euler equations in gas dynamics. Numerical experiments show that for weakly dissipative schemes, slowly moving shock waves fail to transmit the nonconservative interface by producing finally a nonphysical, two-shocked steady-state solution. By using the dynamics of a very slowly moving shock, we will show that two-shocked steady-state solutions are avoided if the interior difference scheme is no less dissipative than the standard Roe scheme even though a nonconservative interface treatment is used.