  The next decade will be an exciting period for solar astrophysics, as new
ground- and space-based instrumentation will provide unprecedented observations
of the solar atmosphere and heliosphere. The synergy between modeling effort
and comprehensive analysis of observations is crucial for the understanding of
the physical processes behind the observed phenomena. However, the
unprecedented wealth of data on one hand, and the complexity of the physical
phenomena on the other, require the development of new approaches in both data
analysis and numerical modeling. In this white paper, we summarize recent
numerical achievements to reproduce structure, dynamics, and observed phenomena
from the photosphere to the low corona and outline challenges we expect to face
for the interpretation of future observations.
