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tokens: imposed in each layer and the wide and position required for the transition region inside the domain. In this simulation we take a 0 = 4.9 × 10 5 K, a 1 = 2 × 10 8 cm, a 2 = 0.5 × 10 8 cm, and a 3 = 5. × 10 5 K. In Fig. 4 the temperature profile obtained for these values and the consequent hydrostatic pressure distribution are shown, for ρ 0 = 10 −11 g cm −3 at the base domain, constant gravity g = 2.74 × 10 4 cm s 2 , specific heat ratio γ = 5/3, and particular gas constant R = 16.6289196 × 10 7 erg (K mol) −1 (totally ionized hydrogen). The numerical results of the hydrostatic condition shown in Fig. 4 are presented in Fig. 5 , where the convergence curves for both the linear and the local hydrostatic reconstruction MUSCL-Hancock schemes are plotted. As expected, the results of the scheme with linear reconstruction exhibit the aforementioned third order superconvergence but they present strong errors in the hydrostatic equilibrium preservation when low resolution grids are used. On the other hand, the results corresponding to the local hydrostatic reconstruction technique show a similar convergence rate for large numbers of cells ( M ≳ 100), but with smaller errors than the standard scheme, which can be even several orders of magnitude smaller when coarse grids are used. This behavior shows the potential of the local hydrostatic reconstruction scheme, since it allows for improvement of the results for very stratified atmospheres with large temperature gradients, as found in many solar astrophysical applications. Fig. 4. Temperature distribution and hydrostatic pressure profile in the solar atmosphere model by Vernazza et al. (1981) given in Eq. ( 63 ). Fig. 5. Convergence curves for the numerical results of the hydrostatic equilibrium atmosphere shown in Fig. 4 . The blue line with circles shows the L 1 -norm of the pressure error for the standard second order MUSCL-Hancock scheme (linear reconstruction), and the red line with crosses corresponds to the MUSCL-Hancock scheme with local hydrostatic reconstruction. 6.2. Diffusion analysis In this analysis we simply study the ability of the local hydrostatic reconstruction scheme to capture a shock wave traveling in a gravitationally stratified medium. The aim of the analysis is to verify that this scheme does not introduce an excessive numerical diffusion compared to the scheme with linear reconstruction to model the shock wave. In addition, we check the behavior of the hydrostatic scheme in the default AMR module of FLASH, which uses the PARAMESH 2 package ( MacNeice et al. 2000 ). This toolkit builds a hierarchy of sub-grids blocks with identical logical structure that cover the physical domain with different resolution levels according to the gradient magnitude of the set refinement variables. Each block is treated as an independent uniform grid, so that the local hydrostatic reconstruction is directly applied on adaptive meshes without the need to modify any subroutine. For this simulation