  Axisymmetric incompressible modes of the magneto-rotational instability (MRI)
with a vertical wavenumber are exact solutions of the non-linear local
equations of motion for a disk (shearing box). They are referred to as "channel
solutions". Here, we generalize a class of these solutions to include energy
losses, viscous, and resistive effects. In the limit of zero shear, we recover
the result that torsional Alfv\'en waves are exact solutions of the non-linear
equations. Our method allows the extension of these solutions into the
dissipative regime.
  These new solutions serve as benchmarks for simulations including dissipation
and energy loss, and to calibrate numerical viscosity and resistivity in the
Zeus3D code. We quantify the anisotropy of numerical dissipation and compute
its scaling with time and space resolution. We find a strong dependence of the
dissipation on the mean magnetic field that may affect the saturation state of
the MRI as computed with Zeus3D. It is also shown that elongated grid cells
generally preclude isotropic dissipation and that a Courant time step smaller
than that which is commonly used should be taken to avoid spurious
anti-diffusion of magnetic field.
