Magnetohydrodynamics (MHD) is the fluid dynamics of conducting fluid or plasma, coupled with Maxwell's equations. The fluid motion induces currents, which produce Lorentz body forces on the fluid. Ampere's Law relates the currents to the magnetic field. The MHD approximation is that the electric field vanishes in the moving fluid frame, except for possible resistive effects.
MHD is described by higher order system of partial differential equations than fluid dynamics. It admits additional waves, the Alfvén waves and their instabilities. A typical feature of MHD is the tendency to form a singular current density. Current sheets, in the presence of resistive dissipation, are associated with the breaking and reconnecting of magnetic field lines.
Various approaches have been used in computational MHD to reduce numerical magnetic dissipation. Lagrangian and partially Lagrangian methods [3] are less diffusive than Eulerian methods, but require substantial rezoning for sheared or straining flow. Mixed finite difference and spectral discretizations have been very effective in dealing with reconnection in periodic geometry [4]. Finite difference codes with nonuniform, Cartesian product grids have also been successful in reconnection simulations in which intense current sheets are aligned with the grid [1]. These approaches, while effective in specialized geometric configurations, are not able to adaptively add grid refinement when current sheets form at arbitrary locations on the mesh. Recently, adaptive structured mesh methods have been successfully applied to these problems [11]. However, these methods are not particularly effective with arbitrarily shaped boundaries. For these reasons we are led to try adaptive unstructured mesh methods. Unstructured meshes have also been used in other computational MHD approaches [10], [12]. We have chosen to apply these methods to two dimensional, incompressible MHD, using a stream function approach to enforce the divergence free conditions on the magnetic and velocity fields. Our approach differs in the use of a symmetrized form of the equations to eliminate difficulties with the calculation of the current density.
The incompressible MHD equations are:
where 
is the magnetic field, 
is the
velocity, and 
is the viscosity.
To enforce incompressibility, it is common to introduce
stream functions:
In two dimensions, with incompressible flow, the MHD equations can be written
where the two dimensional Laplacian is
and
The left hand side of
(7), along with (9) is the familiar
vorticity - stream function formulation
of two dimensional incompressible hydrodynamics. The right hand
side of (7) comes from the Lorentz force
with current density C,, and
the viscosity with coefficient .
Eq. (8) is from Ohm's Law and Faraday's Law and
represents the conservation of magnetic flux 
The z component of the magnetic field does not enter
(7) - (10), so its evolution is
not followed. A large, nearly constant 
is often invoked
to justify the incompressible approximation.
The MHD equations conserve energy and magnetic flux. Since the magnetic
flux function is advected with the flow, any function of 
is a constant of the motion. The energy, 
can be shown
to be conserved by pre multiplying the 
evolution equation by
and the C evolution equation by 
and integrating by
parts, or working directly from the primitive form of the
equations. One obtains
where
assuming either Dirichlet boundary conditions, with 
constant on the boundary, Neumann conditions with the normal
derivatives of equal to zero, or periodic boundary
conditions.
