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

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.


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
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