Semi-implicit characteristic-based boundary treatment for acoustics in low Mach number flows

For low Mach number flow calculation, when acoustic waves have to be captured, semi-implicit methods allow to avoid the time-step limitation that arises when explicit schemes are used. A method is suggested to solve the boundary equations so that the semi-implicitness of the algorithm is maintained, as well as its pressure–velocity coupling. This method is studied theoretically and numerically, in the low Mach number regime. Partially non-reflective characteristic-based boundary conditions, with the linear relaxation form suggested by Rudy and Strikwerda, [J. Comput. Phys. 36 (1980) 55–70], are considered. It is shown that their properties, well known in the framework of explicit schemes, are recovered with the proposed semi-implicit treatment and an acoustic CFL number significantly larger than unity.