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 f_𝐤 is taken towards local equilibrium f_0𝐤, by replacing f_𝐤 - f_0𝐤 via f_𝐤 - f_ S𝐤. The Shakhov-like distribution f_ S𝐤 is constructed using f_0𝐤 and the irreducible moments ρ_r^μ_1 ⋯μ_ℓ of f_𝐤 and reduces to f_0𝐤 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 ℓ_Vπ, ℓ_π 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.