Conclusions

Using rather difficult plane parallel test problems, we have shown that our 3D full-characteristics method gives very good results when compared to our well-tested 1D code. The periodic boundary conditions method discussed here is particularly well-suited to 3-D hydrodynamical simulations of convection in stellar atmospheres and in future work we will compare our results to observations as well as to previous calculations. The results for the computed 3D structure show that $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ leads also to good convergence for a true 3D structure, with convergence rates that are comparable to the simple test cases (see also Papers I and II).