The effects of non-physical mixing on interface development are still not reasonably evaluated when diffuse interface methods (DIMs) are employed to treat the two-medium flows with immiscible interfaces, especially for compressible multi-medium flows with geometrical source terms. In this work, we simulate radially symmetric multi-medium flows employing the sharp interface methods (SIMs) and DIMs to evaluate their performance such as pressure dislocations, mass conservation, and convergence. The $\gamma$-based model and the five-equation transport model are two common DIMs, which are extended to equations with geometrical source terms combined with discontinuous Galerkin (DG) methods. For the SIMs, we employ the classical modified ghost fluid method (MGFM) and its second-order extension (2nd-MGFM) developed recently. Numerical results exhibit that the 2nd-MGFM is more effective in maintaining the interfacial pressure equilibrium relative to the MGFM. The DIMs can always maintain the pressure continuity naturally and total mass conservation, which is not available for SIMs. Further, under the premise of immiscible interfaces defined artificially, the DIMs cannot provide satisfactory single medium mass conservation, while the SIMs have a smaller mass error. In addition, compared to the other three methods, the 2nd-MGFM has higher confidence for radially symmetric flows, matching the exact solution in sparser grids.
