Communications in Mathematical Sciences

Volume 8 (2010)

Number 1

Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part I

Stable discretization of magnetohydrodynamics in bounded domains

Pages: 235 – 251



Jian-Guo Liu

Robert Pego


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.


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

2010 Mathematics Subject Classification

76D03, 76W05

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