Development and astrophysical application of a well-balanced asymptotic preserving scheme for ideal magnetohydrodynamics with gravity

• Applicants: Professor Christian Klingenberg, Ph.D.; Professor Dr. Friedrich Röpke
• Subject Area: Mathematics ; Astrophysics and Astronomy
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 418833237

• Final Report Year: 2024

Final Report Abstract

This project arises from an established cooperation of a mathematics group at Würzburg University (Christian Klingenberg) and an astrophysics group at Heidelberg University (Friedrich Röpke). We aim to develop a new numerical method for ideal magnetohydrodynamics (MHD) with gravity which is well-balanced and asymptotic preserving. This means that the method is able to maintain (magneto-)hydrostatic states exactly and correctly reproduce relevant limits of MHD with gravity, i.e. different low Mach number limits. These properties are important for several stellar astrophysical applications: The well-balanced property can maintain the basic structure of the star while the asymptotic preserving property enables the resolution of slow fluid flows compared to the speed of magnetosonic and Alfvén waves. The new scheme will then be implemented in an existing three-dimensional astrophysical finite volume code. The method will enable us to examine the following three-dimensional astrophysical problems numerically: Firstly, we will examine the effect of magnetic fields on dynamical shear instabilities in differentially rotating stars. Secondly, we are interested in the magnetic field configuration in the onset of core-collapse supernovae.

Publications

• Second Order Finite Volume Scheme for Euler Equations with Gravity which is Well-Balanced for General Equations of State and Grid Systems. Communications in Computational Physics, 26(2), 599-630.
  Berberich, Jonas P.; Chandrashekar, Praveen; null, Christian Klingenberg & Röpke, Friedrich K.
  
• Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers. Astronomy & Astrophysics, 652, A53.
  Edelmann, P. V. F.; Horst, L.; Berberich, J. P.; Andrassy, R.; Higl, J.; Leidi, G.; Klingenberg, C. & Röpke, F. K.
  
• A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors. Astronomy & Astrophysics, 668, A143.
  Leidi, G.; Birke, C.; Andrassy, R.; Higl, J.; Edelmann, P. V. F.; Wiest, G.; Klingenberg, C. & Röpke, F. K.
  
• Dynamics in a stellar convective layer and at its boundary: Comparison of five 3D hydrodynamics codes. Astronomy & Astrophysics, 659, A193.
  Andrassy, R.; Higl, J.; Mao, H.; Mocák, M.; Vlaykov, D. G.; Arnett, W. D.; Baraffe, I.; Campbell, S. W.; Constantino, T.; Edelmann, P. V. F.; Goffrey, T.; Guillet, T.; Herwig, F.; Hirschi, R.; Horst, L.; Leidi, G.; Meakin, C.; Pratt, J.; Rizzuti, F. ... & Woodward, P.
  
• A High Order Semi-implicit Scheme for Ideal Magnetohydrodynamics. Springer Proceedings in Mathematics & Statistics, 21-37. Springer Nature Switzerland.
  Birke, Claudius; Boscheri, Walter & Klingenberg, Christian
  
• A Low Mach Number Two-Speed Relaxation Scheme for Ideal MHD Equations. Springer Proceedings in Mathematics & Statistics, 43-51. Springer Nature Switzerland.
  Birke, Claudius & Klingenberg, Christian
  
• A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity. Communications in Mathematical Sciences, 21(8), 2213-2246.
  Birke, Claudius; Chalons, Christophe & Klingenberg, Christian
  
• Turbulent dynamo action and its effects on the mixing at the convective boundary of an idealized oxygen-burning shell. Astronomy & Astrophysics, 679, A132.
  Leidi, G.; Andrassy, R.; Higl, J.; Edelmann, P. V. F. & Röpke, F. K.
  
• A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers. Journal of Scientific Computing, 98(2).
  Birke, Claudius; Boscheri, Walter & Klingenberg, Christian
  
• Finding an Approximate Riemann Solver via Relaxation: Concept and Advantages. SEMA SIMAI Springer Series, 15-25. Springer Nature Switzerland.
  Birke, Claudius & Klingenberg, Christian
  
• Performance of high-order Godunov-type methods in simulations of astrophysical low Mach number flows. Astronomy & Astrophysics, 686, A34.
  Leidi, G.; Andrassy, R.; Barsukow, W.; Higl, J.; Edelmann, P. V. F. & Röpke, F. K.
  
• Self-consistent magnetohydrodynamic simulation of jet launching in a neutron star – white dwarf merger. Astronomy & Astrophysics, 681, A41.
  Morán-Fraile, Javier; Röpke, Friedrich K.; Pakmor, Rüdiger; Aloy, Miguel A.; Ohlmann, Sebastian T.; Schneider, Fabian R. N.; Leidi, Giovanni & Lioutas, Georgios