As an example and test of the method, we first consider the periodic coalescence instability [1, 2].

The initial equilibrium for the periodic coalescence instability consists of an array of cells with

and We choose the constants The initial equilibrium flux function is shown in Fig.3(a).

We solve the time dependent equations in the domain with periodic boundary conditions. Periodicity is built into the mesh by the connectivity of the mesh triangles. With periodic boundary conditions, coordinate differences are calculated modulo to obtain the mesh - dependent mass and stiffness matrices.

The initial equilibrium is unstable to small perturbations.
Simulations were initialized with arbitrary velocity perturbations,
from which the
unstable mode grows as
The growth rate was extracted by monitoring the kinetic energy as
a function of time. Runs were made advancing both the potential
and current advection forms of the equations, for several
initial mesh sizes. No adaptation was done for these linear
computations. Fig.2 shows the growth rate
as a function of number of mesh points *N*. The two
upper curves show for zero viscosity, using the
two forms of the equations. The curve made with the
current form is marked with *x*'s, while the curve made with
the potential advance using the modified Laplacian
is marked with *o*'s.
There is little difference in the results, which appear to
asymptote to the same linear growth rate. Also shown is
a dashed line, which is
the growth rate with a viscous term, with
corresponding to the value of in finite difference
viscous simulations
[2]. The linear growth rate
is in excellent agreement with the viscous finite difference
simulations. For a mesh of size the
finite difference growth rate Here, the growth
rate for *N* = 10,000 is (by interpolation)

We now consider an adaptive computation of the coalescence
instability. A small viscosity of is used.
We use
the current - vorticity advection formulation of the MHD equations.
Starting with an initial grid of *N* = 2500 points,
the code evolves the equations and refines the mesh.
The flux function at time *t* = 0.21 is shown in
Fig.3(b). The contours of have the form of
cells divided by a nearly pentagonal separatrix. In [1]
it was shown that there is an equilibrium, with pentagonal
separatrices, which has a singular current density along
the shortest side of each pentagon. This equilibrium has
lower energy, and conserves magnetic
flux, relative to the initial state. The singular
equilibrium might be expected as the final state of the time
dependent evolution.

As the simulation evolves, the current density becomes concentrated into
thin sheets located at the short side of the pentagonal separatrix.
A blowup of the plot of the current density at time *t* = 0.21 is
shown in Fig.4(a). The current is well resolved
and unremarkable in structure. A similar blowup of the mesh
on which the current is calculated is shown in
Fig.4(b). The minimum length scale of the
mesh is .022 the length of the original mesh cells, which
is equivalent to a mesh of 5,000,000 mesh points. In fact
the mesh has only 10,400 mesh points.

The peak value of the current
density grows exponentially in time, with a large
growth rate more than 10 times the linear mode growth
rate. This can be seen in Fig.5(a), which shows
the logarithm of the peak current density as a function of time.
The logarithm of the peak current density grows approximately
linearly.
Exponential growth is predicted by a simple linear
model [6].
As the current density increases, so
does the number *N* of mesh points, shown in Fig.5(b).

Wed Jan 7 14:07:46 EST 1998