Magnetic flux rope models and data-driven magnetohydrodynamic simulations of solar eruptions

Solar eruptive activities, such as flares, coronal mass ejections (CMEs), and prominence or filament eruptions, pose both scientific and practical challenges to human beings. To understand and predict these phenomena in the future, we have to combine observations, theoretical, and numerical models closely. Data-driven and data-constrained magnetohydrodynamic (MHD) simulations provide a promising tool to utilize observational data and incorporate relevant physics in the solar atmosphere. They employ magnetic field, along with density and temperature observations, as the initial and boundary conditions. Since there are still difficulties in observing the coronal magnetic field routinely and accurately, we have to rely on magnetic field models in most cases. We introduce some frequently used flux rope models, including the Gibson-Low, Titov-D & eacute;moulin, regularized Biot-Savart laws, nonlinear force-free field, and magnetohydrostatic models. In some studies, the static flux rope models constrained by multi-wavelength observations are used. We discuss many suggested techniques for setting the initial and boundary conditions from both observation and theory, and recent successful applications of the data-driven and data-constrained MHD simulations. We find that data-driven and data-constrained MHD simulations could provide new insights into the physical mechanisms of flux rope formation and eruption, CME structure, and MHD waves.