Level-by-level artificial viscosity and visualization for MHD simulation with adaptive mesh refinement

We propose a numerical method to determine the artificial viscosity in magnetohydrodynamics (MHD) simulations with adaptive mesh refinement (AMR) method, where the artificial viscosity is adaptively changed due to the resolution level of the AMR hierarchy. Although the suitable value of the artificial viscosity depends on the governing equations and the model of target problem, it can be determined by von Neumann stability analysis. By means of the new method, "level-by-level artificial viscosity method," MHD simulations of Rayleigh-Taylor instability (RTI) are carried out with the AMR method. The validity of the level-by-level artificial viscosity method is confirmed by the comparison of the linear growth rates of RTI between the AMR simulations and the simple simulations with uniform grid and uniform artificial viscosity whose resolution is the same as that in the highest level of the AMR simulation. Moreover, in the nonlinear phase of RTI, the secondary instability is clearly observed where the hierarchical data structure of AMR calculation is visualized as high resolution region floats up like terraced fields. In the applications of the method to general fluid simulations, the growth of small structures can be sufficiently reproduced, while the divergence of numerical solutions can be suppressed.