Communications in Mathematical Sciences
Volume 8 (2010)
Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part I
Stable discretization of magnetohydrodynamics in bounded domains
Pages: 235 – 251
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