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Symmetrization of Equations

The evolution of the magnetic and velocity fields are treated in a non symmetric way in the standard formulation above. The velocity is advanced through the vorticity, while the magnetic field is advanced via the magnetic potential. This can cause numerical problems when the equations are solved on an irregular mesh, such as the unstructured, adaptive grids we will use later. It is desirable to formulate the equations in a more symmetrical manner, in which which the current and vorticity are time advanced.

Instead of solving eqs. (8) and (10), we take the Laplacian of (8) and use (9), (5), and (6). This yields an equation for the current, analogous to (7) for the vorticity,



Eqs. (12), (13) are solved along with eqs. (7) and (9). The equations are now symmetrical, in the sense that the source functions tex2html_wrap_inline985 and C are time advanced, and the potentials tex2html_wrap_inline1003 and tex2html_wrap_inline921 are obtained at each time step by solving Poisson equations (9) and (13). Dirichlet, Neumann, or periodic boundary equations are applied to the potentials. With the finite element discretization, no boundary conditions need to be specified for the current and vorticity; the advection equations also hold on the boundary.

The equations can be given an even more symmetric form using the Elsässer variables tex2html_wrap_inline1007 :



Combining the evolution equations for tex2html_wrap_inline985 and C gives (dropping the viscous term)



Hank Strauss
Wed Jan 7 14:07:46 EST 1998