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## A Semi-Implicit Numerical Method for Differentially Rotating Compressible Flows

In astrophysical fluid dynamics, some types of flows, like e.g., magnetorotational super- nova explosions, deal with a highly variable Mach number (from M ≪ 1 in the protoneutron star region to M ≫ 1 near the shock wave expelling from the stellar envelope) in presence of a strong diﬀerential rotation. The Courant–Friedrichs–Lewy (CFL) stability condition of explicit schemes makes such simulations very expensive, even with adaptively refined grids, due to a very large sound speed in the protoneutron star. In such case, semi-implicit schemes (pressure-based solvers) may give a superior performance. In this work, we represent a semi-implicit finite-volume method for solution of the gas dynamical equations, where the acoustic waves are treated implicitly, relaxing the CFL condition. The scheme is implemented in the spherical and cylindrical geometries on an axially moving grid. The latter feature of the proposed approach permits to use the semi-Lagrangian treatment of rotation. It allows to distinguish the large-scale diﬀerential rotation from the scheme, like in approaches suited for accretion discs, decreasing the numerical viscosity, and increasing the stability of the method. Several test calculations demonstrate, that the method is well-suited for eﬀective simulations of both highly compressible and nearly incompressible fluid flows.