Publication 08-CNA-018
**Stable Discretization of Magnetohydrodynamics in Bounded Domains**

Jian-Guo Liu

Department of Mathematics &

Institute for Physical Science and Technology

University of Maryland

College Park, MD 20742

jliu@math.umd.edu

Robert L. Pego

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA 15213

rpego@cmu.edu

**Abstract**: We study a semi-implicit time-difference scheme for
magnetohydrodynamics of a viscous and resistive incompressible fluid in a
bounded smooth domain with perfectly conducting boundary. In the scheme,
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.

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