We perform the lattice simulation to estimate the axion dark matter abundance radiated from the global cosmic strings in the post-inflationary scenario. The independent numerical confirmation on the recently observed logarithmic growth in both the number of strings per Hubble patch and the spectral index of the power-law scaling for the axion spectrum is reported. These logarithmic scalings are checked against two different prescriptions for generating initial random field configurations, namely fat-string type and thermal phase transition. We discuss a possible strong correlation between axion spectrum and the string evolutions with different initial conditions to support the insensitivity of scaling behaviors against different initial data and we provide a qualitative understanding of it. The impact of various combinations of the power law of the axion spectrum, nonlinearities around the QCD scale, and average inter-string distances on the axion abundance is discussed. Additionally, we introduce a new novel string identification method, based on the tetrahedralization of the space, which guarantees the connectedness of the strings and provides a convenient way of assigning the core location. Finally we derive the lower bound on the axion mass.