On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field

0574184 - MÚ 2024 RIV US eng J - Journal Article
Li, Y. - She, Bangwei
On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field.
IMA Journal of Numerical Analysis. Roč. 43, č. 4 (2023), s. 2169-2197. ISSN 0272-4979. E-ISSN 1464-3642
R&D Projects: GA ČR(CZ) GA21-02411S
Institutional support: RVO:67985840
Keywords : compressible MHD * weak-strong uniqueness * stability * convergence
OECD category: Pure mathematics
Impact factor: 2.1, year: 2022
Method of publishing: Limited access
https://doi.org/10.1093/imanum/drac035

We study a general convergence theory for the analysis of numerical solutions to a magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (=2,3). First, we introduce the concept of dissipative weak (DW) solutions and prove the weak-strong uniqueness property for DW solutions, meaning a DW solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the DW solution, as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax equivalence theorem. Finally, as an application of this theory, we show the convergence analysis of two numerical methods.
Permanent Link: https://hdl.handle.net/11104/0344532