A semi-Lagrangian method for the polyatomic ESBGK model
This work provides an efficient and stable numerical method for simulating polyatomic gas flows, which is important for accurately modeling real gases in non-equilibrium conditions.
The authors developed a semi-Lagrangian scheme for the polyatomic ESBGK model that removes time-step restrictions and handles stiff relaxation implicitly, achieving asymptotic preservation to the Euler and Navier-Stokes equations. Simulations of Fourier and Couette flows matched DSMC results, and the method was demonstrated on an orifice flow with moving boundaries.
Polyatomic kinetic models are essential for accurately capturing the thermodynamic behavior of real gases, as internal energy modes significantly influence transport coefficients, relaxation processes, and non-equilibrium effects that cannot be represented by monoatomic models. The polyatomic ESBGK model describes molecular collisions as a relaxation towards a generalized Gaussian distribution with an anisotropic covariance matrix and an exponentially decaying internal energy distribution. We present a new semi-Lagrangian scheme for the polyatomic Ellipsoidal Statistical BGK (ESBGK) model of the Boltzmann equation. The semi-Lagrangian framework, being deterministic and grid-based, removes the time-step restriction associated with the linear transport term by following the method of characteristics. The potentially stiff relaxation term is treated using an implicit A-stable linear multistep method which, owing to the structure of the BGK operator, can be reformulated into a cheap time-stepping scheme. This yields a highly efficient and numerically stable method. The numerical method is asymptotic preserving and stiffly accurate, meaning the scheme asymptotically converges to a scheme for the Euler equations in the vanishing Knudsen limit. In addition, we prove that the first-order scheme, asymptotically converges to the compressible Navier-Stokes equation with correct transport coefficients. Finally, we propose inflow and outflow boundary conditions suitable for BGK-type kinetic equations. We perform simulations of the Fourier and Couette test case to compare the BGK model with Direct Simulation Monte Carlo (DSMC). To conclude, we demonstrate the method on a challenging orifice flow test case with moving boundaries.