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

0562018 - MÚ 2023 RIV US eng J - Journal Article
Li, Y. - She, Bangwei
On convergence of numerical solutions for the compressible MHD system with exactly divergence-free magnetic field.
SIAM Journal on Numerical Analysis. Roč. 60, č. 4 (2022), s. 2182-2202. ISSN 0036-1429. E-ISSN 1095-7170
R&D Projects: GA ČR(CZ) GA21-02411S
Institutional support: RVO:67985840
Keywords : compressible MHD * consistent approximation * convergence * dissipative weak solution
OECD category: Pure mathematics
Impact factor: 2.9, year: 2022
Method of publishing: Limited access
https://doi.org/10.1137/21M1431011

We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence-free property of magnetic field exactly. Our strategy utilizes the recent concepts of dissipative weak solutions and consistent approximations. First, we show the dissipative weak-strong uniqueness principle, meaning a dissipative weak solution coincides with a classical solution as long as they emanate from the same initial data. Next, we show the convergence of consistent approximation toward the dissipative weak solution and thus the classical solution. Upon interpreting the consistent approximation as the stability and consistency of suitable numerical solutions we have established a generalized Lax equivalence theory: convergence - stability and consistency. Further, to illustrate the application of this theory, we propose two mixed finite volume-finite element methods with exact divergence-free magnetic field. Finally, by showing that solutions of these two schemes are consistent approximations, we conclude their convergence toward the dissipative weak solution and the classical solution.
Permanent Link: https://hdl.handle.net/11104/0334447