Axisymmetric flows present unique numerical challenges absent in planar configurations, with bow shocks positioned closer to bodies, compressing stabilizing subsonic layers and promoting carbuncle instabilities. This study conducts the first systematic evaluation of three advection upstream splitting method(AUSM)-family schemes ( Advection Upstream Splitting Method Plus UP(AUSM+UP),Simple Low-dissipation Advection Upstream Splitting method (SLAU), Advection Upstream Splitting Method Plus M (AUSM+M) ) across inviscid and viscous axisymmetric flows spanning transonic to supersonic regimes. Four benchmarks are examined: supersonic sphere (M=4), convergent-divergent nozzle, mixed-compression inlet (M=2), and transonic bump (M=0.875). AUSM+M emerges superior for high- speed axisymmetric applications, uniquely suppressing carbuncle instability on the sphere while achieving 65.8% improvement in post-shock Mach accuracy and 12% shock stand-off error versus AUSM+UP\\\\\\\\\\\\\\\'s 28.5%. For the complex inlet, AUSM+M converges 21.3% faster with optimal total- pressure recovery (3.7% error). It captures shock-induced separation most accurately, predicting separation length within 2.75% on the bump. Conversely, AUSM+UP exhibits severe shock instabilities, while SLAU consistently fails resolving strong shocks despite excellent low-Mach performance. Results establish AUSM+M as the preferred scheme for axisymmetric simulations where shock resolution and numerical stability are paramount.

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