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.
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