Rescaling of the Roe Scheme in Low Mach-Number Flow Regions

Same as Version 1. In preprint format. Some typo errors corrected.A rescaled matrix-valued dissipation is reformulated for the Roe scheme in low Mach-number flow regions from a well known family of local low-speed preconditioners popularized by Turkel. The rescaling is obtained by suppressing the iterative preconditioning and by deriving explicitly the full set of eigenspaces of the Roe-Turkel matrix dissipation. This formulation preserves the time consistency and does not require to reformulate the boundary conditions based on the characteristic theory. The dissipation matrix achieves by construction the proper scaling in low-speed flow regions and returns the original Roe scheme at the sonic line. We find that all eigenvalues are nonnegative in the subsonic regime. By removing the iterative preconditioning, it becomes necessary to formulate a stringent stability condition to the explicit scheme in the low-speed flow regions based on the spectral radius of the rescaled matrix dissipation. This formulation also raises a two-timescale problem for the acoustic waves, which is circumvented for a steady-state iterative procedure by the development of a robust implicit characteristic matrix time-stepping scheme. The behaviour of the modified eigenvalues in the incompressible limit and at the sonic line also suggest applying the entropy correction carefully for complex non-linear flows