Tuned Finite-Difference Diffusion Operators

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.