Adaptive Splitting for Magneto-Hydrodynamics in Astrophysics

Stars such as our Sun feature highly dynamical processes at their surface while their interior is characterized by exceptionally stable physical mechanisms such as the energy generation through fusion of hydrogen. To understand the physics of these objects requires the mathematical modelling of their flow dynamics, of the transport of energy by radiation, and the interaction of the flow with the magnetic fields generated in the stellar plasma. Among others this modelling allows predictions of the emitted radiation. This is of key interest to studies of solar and stellar physics, where continuously growing collections of data have become available through space missions such as the Solar Dynamics Observatory, and new missions such as the Solar Orbiter, and also forecasting solar activity, because the latter can cause failures of telecommunication or electric power grids and also enters climate models. To obtain quantitative information from the mathematical model, we study the propagation in time of the physical quantities described by the basic equations of radiation magneto-hydrodynamics. The underlying models are extremely complex and the equations cannot be solved exactly. Thus, numerical approximations on a computer are employed. Due to the high dimensionality of the equations and the rapid and unstable behavior of their solutions, this requires long computing times. One possibility to reduce the effort is to divide the solution into the consecutive solution of subproblems which describe different physical processes. For the resulting numerical approximation, the deviation from the desired true solution is not known. However, knowledge of this error is crucial, since a predictive model is useful only when the limitations of the numerical results can be estimated. We introduce error estimates which allow efficient and reliable modelling by an adaptive procedure to obtain refined numerical simulations of convective stellar surfaces, also in stars other than the Sun. The results will be compared with observational data, an approach that will benefit from the direct interaction between modelers and observers enabling refinement of the simulations while the computational results will suggest new observing campaigns. The PI O. Koch is an expert in numerical methods and simulation in (quantum) physics, with focus on efficient and precise time integration. He teams up with F. Kupka who is an expert in turbulent convection and stellar astrophysics and with N.J. Mauser as a senior applied mathematician with expertise in numerical modelling of PDE in physics. International collaborators are R. Käppeli, S. Mishra (both ETH Zürich), I. Higueras (Pamplona), and F. Zaussinger (Mittweida).