High-Resolution Finite Element Schemes for the Compressible MHD Equations

• Applicant: Professor Dr. Dmitri Kuzmin
• Subject Area: Mathematics
• Term: from 2014 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 263071379

• Final Report Year: 2019

Final Report Abstract

The development of custom-made limiters and divergence correction techniques for the MHD system was successfully completed by the project participants and their US collaborators. Instead of the splitting-based schemes and a parallel 3D implementation of staggered CT approaches, new artificial viscosity operators, positivity-preserving limiters, and divergence correction procedures were developed for the unstaggered finite element discretization. As of this writing, our FCT scheme appears to be the only numerical method for the ideal MHD equations which guarantees positivity preservation and does not violate conservation laws or maximum principles even in the process of divergence cleaning. The development of this method is the main result of the conducted research and an important milestone in the field of property-preserving algebraic flux correction schemes for general nonlinear systems.

Publications

• (2020) A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations. Journal of Computational Physics 410 109390
  Mabuza, Sibusiso; Shadid, John N.; Cyr, Eric C.; Pawlowski, Roger P.; Kuzmin, Dmitri
  
• (2020) Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations. Journal of Computational Physics 407 109230
  Kuzmin, Dmitri; Klyushnev, Nikita
  
• An FCT finite element scheme for ideal MHD equations in 1D and 2D. J. Comput. Phys. 338 (2017) 585–605
  M. Basting and D. Kuzmin