A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

• Published in: International journal for numerical methods in fluids 2021-12, Vol.93 (12), p.3447-3489
• Main Author: Fambri, Francesco
• Format: Article
• Language: English
• Subjects: Acoustic waves; Algorithms; conservative; Constraints; Convection; Divergence; divergence‐free; Finite volume method; finite‐difference; finite‐volume; Hydrodynamics; Linear algebra; Magnetic field; Magnetic fields; Magnetohydrodynamics; Robustness (mathematics); semi‐implicit; shock‐capturing; Sound waves; Stability; staggered grids; structure‐preserving; Subsystems; three‐split; Velocity; Velocity coupling; Velocity distribution; viscous and resistive MHD
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5041

In this work we introduce a novel semi‐implicit structure‐preserving finite‐volume/finite‐difference scheme for the viscous and resistive equations of magneto‐hydrodynamics (VRMHD) based on an appropriate 3‐split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field, and a third subsystem involving the pressure‐velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfvén waves and the magneto‐acoustic waves are treated implicitly. Thanks to this, the final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence‐free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual (staggered) meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix‐free conjugate gradient algorithm. The final scheme can be regarded as a novel shock‐capturing, conservative, and structure preserving semi‐implicit scheme for VRMHD. Several numerical tests are presented to show the main features of our novel divergence‐free semi‐implicit FV/FD solver: linear‐stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a second‐order of convergence is obtained for a smooth time‐dependent solution; shock‐capturing capabilities are proven against a standard set of stringent MHD shock‐problems; accuracy and robustness are verified against a non‐trivial set of two‐ and three‐dimensional MHD problems. In this work we introduce a novel semi‐implicit structure‐preserving finite‐volume/finite‐difference scheme for viscous and resistive MHD based on an appropriate 3‐split of the governing PDE system: a first convective subsystem, a second subsystem for the Alfvéenic terms, and a third subsystem for the acoustics. The final algorithm is at least formally constrained only by a mild CFL stability condition depending on the fluid velocity. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix‐free conjugate gradient algorithm.