  Finite-difference simulations of fluid dynamics and magnetohydrodynamics
generally require an explicit diffusion operator, either to maintain stability
by attenuating grid-scale structure, or to implement physical diffusivities
such as viscosity or resistivity. If the goal is stability only, the diffusion
must act at the grid scale, but should affect structure at larger scales as
little as possible. For physical diffusivities the diffusion scale depends on
the problem, and diffusion may act at larger scales as well. Diffusivity
undesirably limits the computational timestep in both cases. We construct tuned
finite-difference diffusion operators that minimally limit the timestep while
acting as desired near the diffusion scale. Such operators reach peak values at
the diffusion scale rather than at the grid scale, but behave as standard
operators at larger scales. We focus on the specific applications of
hyperdiffusivity for numerical stabilization, and high Schmidt and high Prandtl
number simulations where the diffusion scale greatly exceeds the grid scale.
