Convergence Analysis of the Grad's Hermite Approximation to the Boltzmann Equation

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion, which could result in non-converging solutions. For linear kinetic equations, a method for posing stable boundary conditions was recently proposed for (formally) arbitrary order Hermite approximations. In the present work, we study $L^2$-convergence of these stable Hermite approximations, and prove explicit convergence rates under suitable regularity assumptions on the exact solution. We confirm the presented convergence rates through numerical experiments involving the linearised-BGK equation of rarefied gas dynamics

Entropy stable hermite approximations of the Boltzmann equation

State of a gas, for all flow regimes, is accurately described by the Boltzmann equation (BE) which governs the evolution of a phase density functional. Presently, out of all available numerical methods, Direct Simulation Monte Carlo (DSMC) solves the BE with highest fidelity. However, DSMC method is expensive for low Mach number flows which motivates the search for other deterministic methods for solving the BE. An appealing deterministic method is a Galerkin type approach which involves approximating the BE's solution in some finite dimensional space. Present work is devoted to developing one of such Galerkin method which is based upon Grad's expansion along the velocity space and a continuous Galerkin type discretization in the physical space. For low Mach number gas flows, we linearise the BE around a local equilibrium distribution function. To approximate the solution to the linearised BE, we develop a Galerkin method which preserves: (i) mass, momentum and energy conservation, (ii) Galelian invariance and, (iii) L2-entropy stability. Many previous works have discussed techniques to preserve the first two properties and in Grad's expansion both of these properties are preserved due to the structure of Hermite polynomials. In the present work, along with preserving the first two properties, we mainly focus on preserving L2-stability for both the velocity and physical space discretization. Solution to the linearised BE lives on a seven dimensional space: three dimensional velocity space, three dimensional physical space and one dimensional temporal space. To develop our Galerkin method, we start with discretizing the velocity space through Grad's Hermite polynomials and we find that the L2-stability of such a discretization solely depends upon the entropy flux across boundaries. This allows us to ensure entropy stability through a proper design of boundary conditions. We design these boundary conditions for both inflow/outflow and solid-wall boundaries. Next, we equip our velocity space discretization with an entropy stable continuous Galerkin spatial discretization. Since a continuous Galerkin method does not allow for discontinuities across cell boundaries, entropy flux across domain's boundary remains as the only source of entropy growth. Therefore, entropy stability of spatial discretization requires a proper boundary discretization. We use a weak boundary discretization to attain entropy stability and we present three different ways of doing so. We compare all of these three approaches through numerical experiments. Stability of a numerical scheme does not only provide a robust numerical implementation but it also allows for an a-priori convergence analysis. We conduct an a-priori convergence analysis for Grad's Hermite expansion where we use its stability to develop error bounds. Under regularity assumptions on linearised BE's solution we develop explicit convergence rates. We confirm the presented convergence rates through numerical experiments involving several benchmark problems. Solving a kinetic equation with a deterministic method is computationally expensive. This motivates developing a deterministic method where both the velocity space and spatial discretization adapts such that computational resources are allocated only where they are needed. We develop such a deterministic method with the help of moment approximations where we change the order of moment approximation and the spatial grid resolution, locally. We focus on steady-state problems and we use goal oriented adjoint based a-posteriori error prediction. With the help of numerical experiments, we compare the convergence behaviour of an adaptive and a uniform deterministic method