  In ergodic quantum spin chains, locally conserved quantities such as energy
or particle number generically evolve according to hydrodynamic equations as
they relax to equilibrium. We investigate the complexity of simulating
hydrodynamics at infinite temperature with multiple methods: time evolving
block decimation (TEBD), TEBD with density matrix truncation (DMT), the
recursion method with a universal operator growth hypothesis (R-UOG), and
operator-size truncated (OST) dynamics. Density matrix truncation and the OST
dynamics give consistent dynamical correlations to $t = 60/J$; and diffusion
constants agreeing within 1%. TEBD only converges for $t\lesssim 20$, but still
produces diffusion coefficients accurate within 1%. The universal operator
growth hypothesis fails to converge and only matches other methods on short
times. We see no evidence of long-time tails in either DMT or OST dynamics
simulations. At finite wavelength, we observe a crossover from purely
diffusive, overdamped decay of the energy density, to underdamped oscillatory
behavior similar to that of cold atom experiments. We dub this behavior "hot
band second sound", and offer a microscopically-motivated toy model.
