Stable discretization of magnetohydrodynamics in bounded domains. (English) Zbl 1278.76127

Liu, Jian-Guo; Pego, Robert

Commun. Math. Sci. 8, No. 1, 235-251 (2010).

Summary: We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently developed formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well-posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

MSC:

76W05	Magnetohydrodynamics and electrohydrodynamics
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M20	Finite difference methods applied to problems in fluid mechanics

Keywords:

time-dependent incompressible viscous flow; Stokes pressure; Leray projection; projection method; pressure Poisson equation