arXiv:2609.17049v2 Announce Type: replace-cross Abstract: Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the intermediate states. We propose a new strategy to design multi-state HLL-type solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, whose compressibility is estimated based on the relations between fast, Alfv\'en, and slow speeds. Then, following the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfv\'en, and slow waves, we calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.