A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows

An immersed discontinuous Galerkin method for compressible Navier-Stokes equations on unstructured meshes

We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh the fluid domain boundary is represented with an implicit signed distance function. The cut-elements partially covered by the solid domain are integrated after tessellation with the marching triangle or tetrahedra algorithms. Two alternative techniques are introduced to overcome the excessive stable time step restrictions imposed by cut-elements. In the first approach the cut-basis functions are replaced with the extrapolated basis functions from the nearest largest element. In the second approach the cut-basis functions are simply scaled proportionally to the fraction of the cut-element covered by the solid. To achieve high-order accuracy additional nodes are introduced on the element faces abutting the solid boundary. Subsequently, the faces are curved by projecting the introduced nodes to the boundary. The proposed approach is verified and validated with several two- and three-dimensional subsonic and hypersonic low Reynolds number flow applications, including the flow over a cylinder, a space capsule and an aerospace vehicle

An immersed discontinuous Galerkin method for compressible Navier-Stokes equations on unstructured meshes

We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh the fluid domain boundary is represented with an implicit signed distance function. The cut-elements partially covered by the solid domain are integrated after tessellation with the marching triangle or tetrahedra algorithms. Two alternative techniques are introduced to overcome the excessive stable time step restrictions imposed by cut-elements. In the first approach the cut-basis functions are replaced with the extrapolated basis functions from the nearest largest element. In the second approach the cut-basis functions are simply scaled proportionally to the fraction of the cut-element covered by the solid. To achieve high-order accuracy additional nodes are introduced on the element faces abutting the solid boundary. Subsequently, the faces are curved by projecting the introduced nodes to the boundary. The proposed approach is verified and validated with several two- and three-dimensional subsonic and hypersonic low Reynolds number flow applications, including the flow over a cylinder, a space capsule and an aerospace vehicle

A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations

In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin (DG) method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. Many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique.Comment: 27 pages, 38 figure

Finite element solution of the Boltzmann equation for rarefied macroscopic gas flows.

This thesis presents research carried out at The Civil and Computational Engineering Centre at Swansea University between September 2004 and December 2007. The focus of the research was the application of modern finite element solution techniques to the governing equations of molecular gas dynamics in order to solve macroscopic gas flow problems. The journey of research began by considering and comparing various finite difference and finite element formulations in the solution of a simple scalar convection equation. This formed the basis for developing a solver for a variety of forms of the Boltzmann equation of molecular gas dynamics, and application of these solvers to a range of subsonic, transonic and supersonic gas flow problems. The merits and drawbacks of the molecular approach, particularly when compared with more traditional continuum CFD solvers, are identified along with possible extensions to the work presented here

General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows

The general synthetic iteration scheme (GSIS) is extended to find the steady-state solution of nonlinear gas kinetic equation, removing the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredients of GSIS are that the gas kinetic equation and macroscopic synthetic equations are tightly coupled, and the constitutive relations in macroscopic synthetic equations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated from the velocity distribution function, however, their constructions are simpler than our previous work (Su et al. Journal of Computational Physics 407 (2020) 109245) for linearized gas kinetic equations. On the other hand, solutions of macroscopic synthetic equations are used to inform the evolution of gas kinetic equation at the next iteration step. A rigorous linear Fourier stability analysis in periodic system shows that the error decay rate of GSIS can be smaller than 0.5, which means that the deviation to steady-state solution can be reduced by 3 orders of magnitude in 10 iterations. Other important advantages of the GSIS are (i) it does not rely on the specific form of Boltzmann collision operator and (ii) it can be solved by sophisticated techniques in computational fluid dynamics, making it amenable to large scale engineering applications. In this paper, the efficiency and accuracy of GSIS is demonstrated by a number of canonical test cases in rarefied gas dynamics.Comment: 25 pages, 17 figures; Version 3, major revision of text and reformed/re-organized equations, added numerical analysis but numerical results are not change