In simulations of rarefied gases, we have to resort to a gas model which goes beyond classical fluid dynamics, as in the presence of shocks the Navier-Stoker-Fourier equations fail to describe the correct behaviour of the thermodynamical quantities. Kinetic methods, which take into account the molecular description of a gas, are a resourceful and accurate alternative. These methods are based on kinetic theory, which states that the thermodynamical quantities of a gas can be described as moments of the molecular velocity distribution $f(v,x,t)$. For example, we can write the density, bulk velocity and energy $\rho, U, E$ as
$$
\rho = \int_{R^3} f \ dv, \ U = \cfrac{1}{\rho} \int_{R^3} v\ f \ dv, \ E = \cfrac{1}{2\rho} \int_{R^3} \vert v -U\vert^2 f \ dv.
$$
The gas dynamics is obtained by following the evolution of $f$ with respect to the binary interaction between the particles (i.e. collisions). This is described by the Boltzmann equation
$$
\partial_t f + v \cdot \nabla_x f = S[f]
$$
where $S[f]$ is called the collision operator. In general, $S(f)$ has a complex integral form and therefore it needs to be modeled in a computationally advantageous way. Common approaches such as Direct Simulation Monte Carlo (DSMC) sample colliding pairs through a Monte Carlo procedure, however this becomes unfeasible when the average number of collisions is high.
In recent years, stochastic approximations of the Boltzmann equation based on simplified collision operators have gained popularity for being a computationally affordable alternative to DSMC. Within these methods, collision operators constructed from the Fokker-Planck equation have been widely studied given their promising potentials.
In fact, the Fokker-Planck equation
$$
\partial_t f + v \cdot \nabla_x f = -\nabla_v \cdot ( A[f] \ f ) + \Delta_{v,v}(D[f] \ f)
$$
gives rise to a drift-diffusion process with drift functional $A[f]$ and diffusion matrix $D[f]$. This leads to a system of Stochastic Differential Equations (SDEs) of Langevin type for the stochastic processes $X_t,V_t$ which describe the position and velocity of the particles:
$$
dX_t = V_t dt, \
dV_t = A[f] dt + \sqrt{2D[f]}dW_t,
$$
where $dW_t$ is the standard Wiener process. This leads to algorithms which are independent of the collision frequency and only depend on the number of particles, which become particularly advantageous in the near-continuum regimes. However, despite this potential, several limitaton in the model still exist. In particular, we want to address the problem of devising a model which honours prescribed moment dynamics along with the entropy constraint. While the former ensures recovery of correct conservation laws, and thus Navier-Stokes-Fourier system in the continuum limit, the latter leads to the model stability for an arbitrary distribution function.
In this talk, we present a Fokker-Planck model which satisfies the aforementioned criteria. In particular, the model includes a novel closure that enforces negative entropy decay proportional to the Fisher information. After having discussed the model, we will present validation of theoretical results. we will conclude by showing the comparison of runtimes against DSMC in a real-world setting with varying rarefaction regimes. Next steps and addressing of remaining open problems will be also outlined.