Quadratic Fokker-Planck model of monatomic rarefied gas

In recent years, significant progress has been made in Fokker-Planck (FP) approximations of the Boltzmann equation, where binary collisions are modeled as drift and diffusion processes in velocity space. To address the discrepancy in the Prandtl number for the original linear FP model, several modified formulations have been developed. Among these, one strategy introduces additional degrees of freedom into the drift term. However, this leads to an increase in model complexity, as the presence of higher-order velocity moments complicates the solution of the equations with underdetermined parameters. An alternative strategy primarily modifies the diffusion term, but the anisotropy introduced in the diffusion term may compromise its positive definiteness. In this work, a quadratic-FP (quad-FP) model with correct Prandtl number is proposed. In the quad-FP model, a quadratic nonlinear modification is introduced into the drift term and corresponding anisotropy modifications are also implemented in the diffusion term. This design simplifies the model structure and facilitates numerical implementation. Moreover, it mitigates the anisotropy introduced by diffusion-term modification, better preserving the positive definiteness of the diffusion term even under strongly nonequilibrium conditions. Numerical tests demonstrate that the quad-FP model provides predictions consistent with direct simulation Monte Carlo results in a variety of flows and achieves improved computational efficiency.