High-order Shakhov-like extension of the relaxation time approximation in relativistic kinetic theory

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 fₖ is taken toward local equilibrium f₀ₖ, by replacing fₖ-f₀ₖ via fₖ-fSk. The Shakhov-like distribution fSk is constructed using f₀ₖ and the irreducible moments ρᵣ^μ₁⋯μ_ of fₖ and reduces to f₀ₖ 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 λ_Ππ in the frame of the Bjorken flow of massive particles, as well as the diffusion-shear transport coefficients _Vπ, _πV 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.