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Core-collapse supernovae span a wide range of energies, from much less thanto much greater than the binding energy of the progenitor star. As a result,the shock wave generated from a supernova explosion can have a wide range ofMach numbers. In this paper, we investigate the propagation of shocks witharbitrary initial strengths in polytropic stellar envelopes using a suite ofspherically symmetric hydrodynamic simulations. We interpret these resultsusing the three known self-similar solutions for this problem: the Sedov-Taylorblastwave describes an infinitely strong shock and the self-similar solutionsfrom Coughlin et al. (2018b) (Paper I) and Coughlin et al. (2019) (Paper II)describe a weak and infinitely weak shock (the latter being a rarefactionwave). We find that shocks, no matter their initial strengths, evolve towardeither the infinitely strong or infinitely weak self-similar solutions atsufficiently late times. For a given density profile, a single functioncharacterizes the long-term evolution of a shock's radius and strength.However, shocks with strengths near the self-similar solution for a weak shock(from Paper I) evolve extremely slowly with time. Therefore, the self-similarsolutions for infinitely strong and infinitely weak shocks are not likely to berealized in low-energy stellar explosions, which will instead retain memory ofthe shock strength initiated in the stellar interior.

Weak Shock Propagation with Accretion. III. A Numerical Study On Shock Propagation & Stability