Communications in Mathematical Sciences

Stable discretization of magnetohydrodynamics in bounded domains

Jian-Guo Liu and Robert Pego

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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 de- veloped 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.

Article information

Commun. Math. Sci., Volume 8, Number 1 (2010), 235-251.

First available in Project Euclid: 23 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76W05: Magnetohydrodynamics and electrohydrodynamics 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

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


Liu, Jian-Guo; Pego, Robert. Stable discretization of magnetohydrodynamics in bounded domains. Commun. Math. Sci. 8 (2010), no. 1, 235--251.

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