  To simulate elastic turbulence, where viscoelasticity dominates, numerical
solvers introduce an artificial stress diffusivity term to handle the steep
polymer stress gradients that ensue. This has recently been shown [Gupta &
Vincenzi, J. Fluid Mech. 870, 405-418 (2019); Dzanic, From & Sauret, J. Fluid
Mech. 937, A31 (2022)] to introduce unphysical artifacts with a detrimental
impact on simulations. In this Letter, we propose that artificial diffusion is
limited to regions where stress gradients are steep instead of seeking the
zero-diffusivity limit. Through the cellular forcing and four-roll mill
problem, we demonstrate that this modified artificial diffusivity is devoid of
unphysical artifacts, allowing all features of elastic turbulence to be
retained. Results are found to conform with direct simulations, reducing the
impact of artificial diffusivity from a qualitative scale to a quantitative
scale while only requiring a fraction of the numerical resolution.
