  We derive the transport coefficients of second-order fluid dynamics with $14$
dynamical moments using the method of moments and the Chapman-Enskog method in
the relaxation-time approximation for the collision integral of the
relativistic Boltzmann equation. Contrary to results previously reported in the
literature, we find that the second-order transport coefficients derived using
the two methods are in perfect agreement. Furthermore, we show that, unlike in
the case of binary hard-sphere interactions, the diffusion-shear coupling
coefficients $\ell_{V\pi}$, $\lambda_{V\pi}$, and $\tau_{V\pi}$ actually
diverge in some approximations when the expansion order $N_\ell \rightarrow
\infty$. Here we show how to circumvent such a problem in multiple ways,
recovering the correct transport coefficients of second-order fluid dynamics
with $14$ dynamical moments. We also validate our results for the
diffusion-shear coupling by comparison to a numerical solution of the Boltzmann
equation for the propagation of sound waves in an ultrarelativistic ideal gas.
