Aims. We present the first high-resolution non-equilibrium ionization simulation of the joint evolution of the Local Bubble (LB) and Loop I superbubbles in the turbulent supernova-driven interstellar medium (ISM). The time variation and spatial distribution of the Li-like ions Civ, Nv, and Ovi inside the LB are studied in detail. Methods. This work uses the parallel adaptive mesh refinement code EAF-PAMR coupled to the newly developed atomic and molecular plasma emission module E(A+M)PEC, featuring the time-dependent calculation of the ionization structure of H through Fe, using the latest revision of solar abundances. The finest AMR resolution is 1 pc within a grid that covers a representative patch of the Galactic disk (with an area of 1 kpc^2 in the midplane) and halo (extending up to 10 kpc above and below the midplane). Results. The evolution age of the LB is derived by the match between the simulated and observed absorption features of the Li-like ions Civ, Nv, and Ovi . The modeled LB current evolution time is bracketed between 0.5 and 0.8 Myr since the last supernova reheated the cavity in order to have N(Ovi) < 8 times 10^12 cm-2, log[N(Civ) /N(Ovi) ] < -0.9 and log[N(Nv) /N(Ovi) ] < -1 inside the simulated LB cavity, as found in Copernicus, IUE, GHRS-IST and FUSE observations.
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