In this paper we present a relativistic Shakhov-type generalization of the Anderson-Witting relaxation time model for the Boltzmann collision integral. The extension is performed by modifying the path on which the distribution function $f_{ \mathbf{ k}}$ is taken towards local equilibrium $f_{ 0\mathbf{ k}}$, by replacing $f_{ \mathbf{ k}} - f_{ 0\mathbf{ k}}$ via $f_{ \mathbf{ k}} - f_{ { \rm S}\mathbf{ k}}$. The Shakhov-like distribution $f_{ { \rm S} \mathbf{ k}}$ is constructed using $f_{ 0\mathbf{ k}}$ and the irreducible moments $\rho_r^{ \mu_1 \cdots \mu_\ell}$ of $f_\mathbf{ k}$ and reduces to $f_{ 0\mathbf{ k}}$ in local equilibrium. Employing the method of moments, we derive systematic high-order Shakhov extensions that allow both the first- and the second-order transport coefficients to be controlled independently of each other. We illustrate the capabilities of the formalism by tweaking the shear-bulk coupling coefficient $\lambda_{ \Pi \pi}$ in the frame of the Bjorken flow of massive particles, as well as the diffusion-shear transport coefficients $\ell_{ V\pi}$, $\ell_{ \pi V}$ in the frame of sound wave propagation in an ultrarelativistic gas. Finally, we illustrate the importance of second-order transport coefficients by comparison with the results of the stochastic BAMPS method in the context of the one-dimensional Riemann problem.
Categories: nucl-th physics.flu-dyn