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We develop the steady-state regularized 13-moment equations in the linear regime for rarefied gas dynamics with general collision models. For small Knudsen numbers, the model is accurate up to the super-Burnett order, and the resulting system of moment equations is shown to have a symmetric structure. We also propose Onsager boundary conditions for the moment equations that guarantee the stability of the equations. The validity of our model is verified by benchmark examples for the one-dimensional channel flows. [ABSTRACT FROM AUTHOR]

A comparative investigation of the hydrodynamic instability development on the shock-driven square and rectangular light gas bubbles is carried out numerically. In contrast to the square bubble, both horizontally and vertically aligned rectangular bubbles with different aspect ratios are taken into consideration, highlighting the impacts of aspect ratios on interface morphology, vorticity production, and bubble deformation. Two-dimensional compressible Euler equations for two-component gas flows are simulated with a high-order modal discontinuous Galerkin solver. The results show that the aspect ratio of rectangular bubbles has a considerable impact on the evolution of interface morphology in comparison with a square bubble. In horizontal-aligned rectangular bubbles, two secondary vortex rings connected to the primary vortex ring are produced by raising the aspect ratio. While in vertical-aligned rectangular bubbles, two re-entrant jets are seen close to the top and bottom boundaries of the upstream interface with increasing aspect ratio. The baroclinic vorticity generation affects the deformation of the bubble interface and accelerates the turbulent mixing. Notably, the complexity of the vorticity field keeps growing as the aspect ratio does in horizontal-aligned rectangular bubbles, and the trends are reversed in the vertical-aligned rectangular bubbles. Further, these aspect ratio effects also lead to the different mechanisms of the interface characteristics, including the upstream and downstream distances, width, and height. Finally, the temporal evolution of spatially integrated fields, including average vorticity, vorticity production terms, and enstrophy are analyzed in depth to investigate the impact of aspect ratio on the flow structure. [ABSTRACT FROM AUTHOR]

For real-world engineering applications, there is a great deal of interest in developing effective models and simulations of continuum-rarefied gas flows. In this study, the numerical simulations of Grad's 13 (G13) moment equations with application to the Riemann problem in a wide range of continuum-rarefied flow regimes are presented. This work emphasizes numerical robustness and wave phenomena in the G13 system to provide a building block for regularized 13 moment systems, and high-order Grad's models. For this purpose, a high-order modal discontinuous Galerkin solver is developed for solving one-dimensional G13 moment equations. For spatial discretization, hierarchical modal basis functions premised on orthogonal-scaled Legendre polynomials are used. The proposed approach reduces the G13 systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order SSP Runge-Kutta algorithm. Three Riemann test cases, including the shock tube, two shock waves and two rarefaction waves, are examined in continuum-rarefied flow regimes. In the G13 system, the arising characteristic waves and dissipation phenomena are investigated in depth. The numerical results demonstrate that every Riemann problem does not have a solution for the G13 system because of loss of hyperbolicity of the system. [ABSTRACT FROM AUTHOR]

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS' Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly. [ABSTRACT FROM AUTHOR]

Moment expansions are used as a model reduction technique in kinetic gas theory to approximate the Boltzmann equation. Rarefied gas models based on so-called moment equations have become increasingly popular recently. However, in a seminal paper by Holway ["Existence of kinetic theory solutions to the shock structure problem," Phys. Fluids 7(6), 911–913 (1965)], a fundamental restriction on the existence of the expansion was used to explain subshock behavior of shock profile solutions obtained by moment equations. Later, Weiss ["Comments on 'Existence of kinetic theory solutions to the shock structure problem' [Phys. Fluids 7, 911 (1964)]," Phys. Fluids 8(6), 1689–1690 (1996)] argued that this restriction does not exist. We will revisit and discuss their findings and explain that both arguments have a correct and an incorrect part. While a general convergence restriction for moment expansions does exist, it cannot be attributed to subshock solutions. We will also discuss the implications of the restriction and give some numerical evidence for our considerations. [ABSTRACT FROM AUTHOR]

To obtain a symmetric hyperbolic moment system from the linearised Boltzmann equation, we approximate the entropy variable (derivative of the entropy functional) with the help of multi-variate polynomials in the velocity space. Choosing the entropy functional to be quadratic, we retrieve the Grad's approximation for the linearised Boltzmann equation. We develop a necessary and sufficient condition for the entropy stability of the Grad's approximation on bounded position domain with inflow and outflow boundaries. These conditions show the importance of using the Onsager Boundary Conditions (OBCs) Rana and Struchtrup (2016) [28] for obtaining entropy stability and we use them to prove that a broad class of Grad's approximations, equipped with boundary conditions obtained through continuity of odd fluxes Grad (1949) [17] , are entropy unstable. Entropy stability, for the Grad's approximation, is obtained through entropy stabilization of the boundary conditions obtained through the continuity of odd fluxes. Since many practical implementations require the prescription of an inflow velocity, the entropy bounds for two possible methods to achieve the same is discussed in detail, both for the linearised Boltzmann equation and its Hermite approximation. We use the Discontinuous Galerkin (DG) discretization in the physical space, to study several benchmark problems to ascertain the physical accuracy of the proposed entropy stable Grad's approximation. [ABSTRACT FROM AUTHOR]

The recent development of the quadrature-based moment equations (QBME) for the simulation of rarefied gases using the Boltzmann equation led to a promising hyperbolic moment model. This paper deals with the two-dimensional QBME model, presents its derivation based on different possible expansions, and gives explicit equations for the QBME model and related models before describing a numerical method to solve the nonconservative PDE system on two-dimensional, unstructured grids. The first simulations using the two-dimensional QBME are shown in this paper using the flow past a cylinder and a forward facing step test case. The results indicate the applicability of the QBME models for rarefied gas flows as well as convergence to the Euler solution in the case of vanishing Knudsen number. [ABSTRACT FROM AUTHOR]

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331-407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments. [ABSTRACT FROM AUTHOR]

We are interested in the numerical solution of large systems of hyperbolic conservation laws or systems in which the characteristic decomposition is expensive to compute. Solving such equations using finite volumes or discontinuous Galerkin requires a numerical flux function which solves local Riemann problems at cell interfaces. There are vario us methods to express the numerical flux function. On the one end, there is the robust but very diffusive Lax--Friedrichs solver; on the other end is the upwind Godunov solver which respects all resulting waves. The drawback of the latter method is the costly computation of the eigensystem. This work presents a family of simple first-order Riemann solvers, named, HLLXω, which avoid solving the eigensystem. The new method reproduces all waves of the system with less dissipation than other solvers with similar input and effort, such as HLL and FORCE. The family of Riemann solvers can be seen as an extension or generalization of the methods introduced by Degond et al. [C.R. Acad. Sci. Paris Ser. I, 328 (1999), pp. 479--483]. We only require the same number of input values as HLL, namely, the globally fastest wave speeds in both directions, or an estimate of the speeds. Thus, the new family of Riemann solvers is particularly efficient for large systems of conservation laws when the spectral decomposition is expensive to compute or no explicit expression for the eigensystem is available. [ABSTRACT FROM AUTHOR]

We present an analysis of a set of parametrized boundary conditions for a Stokes–Brinkman model in two space dimensions, discretized by finite elements. We particularly point out an instability which arises when these boundary conditions are posed on a curved line, which then leads to unphysical oscillations. In contrast to a Navier-slip condition, which is prone to Babuška's paradox, this instability can be traced back to the continuous level. We claim that the stability in these cases depend on the amount of curvature at the boundary, which is shown in a reduced setting in cylinder coordinates. The transition to a two dimensional Cartesian case is then based on numerical studies, which further substantiate the claim. Lastly, stabilization techniques are motivated that enhance the continuous FEM setting and are conveniently able to deal with arising oscillations. [ABSTRACT FROM AUTHOR]

A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles. [ABSTRACT FROM AUTHOR]

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem. [ABSTRACT FROM AUTHOR]

The strategy for computing the Boltzmann collision integrals for gaseous mixtures is presented and bestowed to compute the fully non-linear Boltzmann collision integrals for hard sphere gas-mixtures. The Boltzmann collision integrals associated with the first 26 moments of each constituent in a gas-mixture are presented. Moreover, the Boltzmann collision integrals are exploited to study the relaxation phenomena of diffusion velocities, stresses and heat fluxes in binary gas-mixtures of Maxwell molecules and hard spheres. [ABSTRACT FROM AUTHOR]

This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number-the ratio between the mean free path of the particles and a macroscopic length scale-becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann's equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations. [ABSTRACT FROM AUTHOR]

We present a novel approach to solve the Vlasov–Poisson equation: the numerical flow iteration. It evaluates the numerical solution by storing the low‐dimensional electric potentials and uses these to iteratively reconstruct the characteristics. This reduces the total memory footprint by several orders of magnitude as complexity gets shifted from memory access to computation on‐the‐fly. Furthermore, this ansatz yields strong conservation properties due to the exploitation of the solution structure. [ABSTRACT FROM AUTHOR]

A sequence of approximate linear collision models for hard-sphere and inverse-power-law gases is introduced. These models are obtained by expanding the linearized Boltzmann collision operator into series, and a practical algorithm is proposed for evaluating the coefficients in the series. The sequence is proven to be convergent to the linearized Boltzmann operator, and it established a connection between the Shakhov model and the linearized collision model. The convergence is demonstrated by solving the spatially homogeneous Boltzmann equation. By observing the magnitudes of the coefficients, simpler models are developed through removing small entries in the coefficient matrices. [ABSTRACT FROM AUTHOR]

The strategy for computing the Boltzmann collision integrals for gaseous mixtures is presented and bestowed to compute the fully non-linear Boltzmann collision integrals for hard sphere gas-mixtures. The Boltzmann collision integrals associated with the first 26 moments of each constituent in a gas-mixture are presented. Moreover, the Boltzmann collision integrals are exploited to study the relaxation phenomena of diffusion velocities, stresses and heat fluxes in binary gas-mixtures of Maxwell molecules and hard spheres. [ABSTRACT FROM AUTHOR]

The fully nonlinear Grad's N × 26-moment (N × G26) equations for a mixture of N monatomic-inert-ideal gases made up of Maxwell molecules are derived. The boundary conditions for these equations are derived by using Maxwell's accommodation model for each component in the mixture. The linear stability analysis is performed to show that the 2 × G26 equations for a binary gas mixture of Maxwell molecules are linearly stable. The derived equations are used to study the heat flux problem for binary gas mixtures confined between parallel plates having different temperatures. [ABSTRACT FROM AUTHOR]

Abstract: The use of moment closures for the prediction of continuum and moderately non-equilibrium flows offers modelling and numerical advantages over other methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however their are two issues that limit their practical implementation. Firstly, for closures that have a treatment for heat transfer, fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation. Secondly, for these same closures, there are physically possible moment states for which the entropy-maximization problem has no solution and the entire framework breaks down. This paper demonstrates that affordable closed-form moment closures that are inspired by the maximum-entropy framework can be proposed. It is known that closing fluxes in the maximum-entropy hierarchy approach a singularity as the region of non-solvability is approached. This paper shows that, far from a disadvantage, this singularity allows for smooth and accurate prediction of shock-wave structure, even for high Mach numbers. The presence of unphysical “sub-shocks” within shock-profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment-closure technique. The realization that smooth shock profiles are, in fact, possible for moment methods with a moderate number of moments greatly increases the method’s applicability to high-speed flows. In this paper, a 5-moment system for a simple one-dimensional gas and a 14-moment system for realistic gases are developed and examined. Numerical solution for shock-waves at a variety of incoming flow Mach numbers demonstrate both the robustness and the accuracy of the closures. [Copyright &y& Elsevier]

The regularized 13 moment equations of rarefied gas dynamics are derived for a monatomic hard sphere gas in the linear regime. The equations are based on an extended Grad-type moment system, which is systematically reduced by means of the Order of Magnitude Method [H. Struchtrup, 'Stable transport equations for rarefied gases at high orders in the Knudsen number,' Phys. Fluids 16(11), 3921-3934 (2004)]. Chapman-Enskog expansion of the final equations yields the linear Burnett and super-Burnett equations. While the Burnett coefficients agree with literature values, this seems to be the first time that super-Burnett coefficients are computed for a hard sphere gas. As a first test of the equations the dispersion and damping of sound waves is considered. [ABSTRACT FROM AUTHOR]

Abstract: In this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment (R13) equations of rarefied gas dynamics. The scheme allows fast solutions for 2D and 3D boundary value problems (BVPs) with velocity slip and temperature jump boundary conditions. The scheme is applied to the lid driven cavity problem for Knudsen numbers up to 0.7. The results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Integro Moment Method (IMM) and the Direct Simulation Monte Carlo (DSMC) method. The R13 equations yield better results than the classical Navier–Stokes–Fourier equations for this boundary value problem, since they give an approximate description of Knudsen boundary layers at moderate Knudsen numbers. The R13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate Knudsen numbers. [Copyright &y& Elsevier]

Abstract: In this article, a new model and a biomimetic pattern for heliostat field layout optimization are introduced. The model, described and validated herein, includes a detailed calculation of the annual average optical efficiency accounting for cosine losses, shading and blocking, aberration and atmospheric attenuation. The model is based on a discretization of the heliostats and can be viewed as ray tracing with a carefully selected distribution of rays. The prototype implementation is sufficiently fast to allow for field optimization. Parameters are introduced for the radially staggered layout and are optimized with the objective of maximizing the annual insolation weighted heliostat field efficiency. In addition, inspired by the spirals of the phyllotaxis disc pattern, a new biomimetic placement heuristic is described and evaluated, which generates layouts of both higher insolation-weighted efficiency and higher ground coverage than radially staggered designs. Specifically, this new heuristic is shown to improve the existing PS10 field by 0.36% points in efficiency while simultaneously reducing the land area by 15.8%. Moreover, the new pattern achieves a better trade-off between land area usage and efficiency, i.e., it can reduce the area requirement significantly for any desired efficiency. Finally, the improvement in area becomes more pronounced with an increased number of heliostats, when maximal efficiency is the objective. While minimizing the levelized cost of energy (LCOE) is typically a more practical objective, results of the case study presented show that it is possible to both reduce the land area (i.e. footprint) of the plant and number of heliostats for fixed energy collected. By reducing the capital cost of the plant at no additional costs, the effect is a reduction in LCOE. [Copyright &y& Elsevier]

The order-of-magnitude method proposed by Struchtrup (Phys. Fluids 16(11):3921-3934, ) is a new closure procedure for the infinite moment hierarchy in kinetic theory of gases, taking into account the scaling of the moments. The scaling parameter is the Knudsen number Kn, which is the mean free path of a particle divided by the system size. In this paper, we generalize the order-of-magnitude method and derive a formal theory of scale-induced closures on the level of the kinetic equation. Generally, different orders of magnitude appear through balancing the stiff production term of order 1/ Kn with the advection part of the kinetic equation. A cascade of scales is then induced by different powers of Kn. The new closure produces a moment distribution function that respects the scaling of a Chapman-Enskog expansion. The collision operator induces a decomposition of the non-equilibrium part of the distribution function in terms of the Knudsen number. The first iteration of the new closure can be shown to be of second-order in Kn under moderate conditions on the collision operator, to be L-stable and to possess an entropy law. The derivation of higher order approximations is also possible. We illustrate the features of this approach in the framework of a 16 discrete velocities model. [ABSTRACT FROM AUTHOR]

The regularized 13-moment equations are solved analytically for the microflow of a gas past a sphere in the case of low Mach numbers. The result is given in fully explicit expressions and shows nontrivial behavior for all fluid fields including stress, heat flux, and temperature. Various aspects of the flow such as temperature polarization and total force are reproduced correctly for moderate Knudsen number. The analytical solution allows studying the rise of Knudsen layers and their interaction and coupling to the fluid variables in the bulk. Additionally, based on the regularized 13-moment equations system, hybrid boundary conditions are given for the standard Stokes equations in order to enable them to predict nonequilibrium effects in the flow past a sphere. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale. The stochastic model represents a Fokker–Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime. Numerical studies demonstrate that the stochastic model is consistent with Navier–Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented. [Copyright &y& Elsevier]

Abstract: We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to non-linear smooth functions, we introduce a new class of limiter functions for the spatial reconstruction of hyperbolic equations. We therefore employ and generalize the idea of double-logarithmic reconstruction of Artebrant and Schroll [R. Artebrant, H.J. Schroll, Limiter-free third order logarithmic reconstruction, SIAM J. Sci. Comput. 28 (2006) 359-381]. The result is a class of efficient third-order schemes with a compact three-point stencil. The interface values between two neighboring cells are obtained by a single limiter function. The limiter belongs to a family of functions, which are based upon a non-polynomial and non-linear reconstruction function. The new methods handle discontinuities as well as local extrema within the standard semi-discrete TVD-MUSCL framework using only a local three-point stencil and an explicit TVD Runge–Kutta time-marching scheme. The shape-preserving properties of the reconstruction are significantly improved, resulting in sharp, accurate and symmetric shock capturing. Smearing, clipping and squaring effects of classical second-order limiters are completely avoided. Computational efficiency is enhanced due to large allowable Courant numbers , as indicated by the von Neumann stability analysis. Numerical experiments for a variety of hyperbolic partial differential equations, such as Euler equations and ideal magneto-hydrodynamic equations, confirm a significant improvement of shock resolution, high accuracy for smooth functions and computational efficiency. [Copyright &y& Elsevier]

The regularized 13-moment equations for rarefied gas flows are considered for planar microchannel flows. The governing equations and corresponding kinetic boundary conditions are partly linearized, such that analytical solutions become feasible. The nonlinear terms include contributions of the shear stress and shear rate, which describe the coupling between velocity and temperature fields. Solutions for Couette and force-driven Poiseuille flows show good agreement with direct simulation Monte Carlo data. Typical rarefaction effects, e.g., heat flux parallel to the wall and the characteristic dip in the temperature profile in Poiseuille flow, are reproduced accurately. Furthermore, boundary effects such as velocity slip, temperature jump, and Knudsen boundary layers are predicted correctly. [ABSTRACT FROM AUTHOR]

Abstract: It is of utmost interest to control the divergence of the magnetic flux in simulations of the ideal magnetohydrodynamic equations since, in general, divergence errors tend to accumulate and render the schemes unstable. This paper presents a higher-order extension of the locally divergence-preserving procedure developed in Torrilhon [M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput. 26 (2005) 1166–1191]; a fourth-order accurate local redistribution of the numerical magnetic field fluxes of a finite volume base scheme is introduced. The redistribution ensures that a fourth-order accurate discrete divergence operator is preserved to round off errors when applied to the cell averages of the magnetic flux density. The developed procedure is applicable to generic semi-discrete finite volume schemes and its purpose is to stabilize the schemes using a local procedure that respects the accuracy of the base scheme to a greater extent than the previous second-order achievements. Numerical experiments that demonstrate the properties of the new procedure are also presented. [Copyright &y& Elsevier]

Abstract: Boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro-flows of gases. In this paper, we present a theory how to combine the regularized 13-moment-equations derived from Boltzmann’s equation with boundary conditions obtained from Maxwell’s kinetic accommodation model. While for the linear case these kinetic boundary conditions suffice, we need additional conditions in the non-linear case. These are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to Boltzmann’s equation. After finding a suitable set of boundary conditions and equations, a numerical method for generic shear flow problems is formulated. Several test simulations demonstrate the stable and oscillation-free performance of the new approach. [Copyright &y& Elsevier]

The newly derived fluid model named regularized 13-moment system (R13) is capable of describing rarefied/microflows with high accuracy due to special modeling of the nonequilibrium dissipation. Explicit two-dimensional equations are presented and discussed and a specialized numerical method derived. In two dimensions evolution equations for nine basic variables need to be solved, including directional temperatures, shear stress, and heat flux. The governing partial differential equations are given in balance law form with relaxation and parabolic dissipation. The numerical method is formulated as a practical and straightforward extension of standard finite volume methods. It focuses on bulk simulations with no boundary in order to provide a building block. As a microflow example the interaction of a shock front with a dense microbubble is considered in which the bubble's diameter is a few mean free paths. The results are compared to the solution of the Navier-Stokes-Fourier system demonstrating the relevance of the new R13 model. [ABSTRACT FROM AUTHOR]

A main issue in nonstationary, compressible magnetohydrodynamic (MHD) simulations is controlling the divergence of the magnetic flux. This paper presents a general procedure showing how to modify the intercell fluxes in a conservative MHD finite volume code such that the scheme becomes locally divergence preserving. That is, a certain discrete divergence operator vanishes exactly during the entire simulation, which results in the suppression of any divergence error. The procedure applies to arbitrary finite volume schemes provided they are based on intercell fluxes. We deduce the necessary modifications for numerical methods based on rectangles and triangles and present numerical experiments with the new schemes. The theoretical justifition of the schemes is given in two independent ways. One way starts with the discrete divergence operator that has to be preserved and modifies the fluxes accordingly. The second way uses a finite element reconstruction via Nedelec elements. Both methods lead to equivalent numerical methods. [ABSTRACT FROM AUTHOR]

A new closure for Grad's 13 moment equations is presented that adds terms of Super-Burnett order to the balances of pressure deviator and heat flux vector. The additional terms are derived from equations for higher moments by means of the distribution function for 13 moments. The resulting system of equations contains the Burnett and Super-Burnett equations when expanded in a series in the Knudsen number. However, other than the Burnett and Super-Burnett equations, the new set of equations is linearly stable for all wavelengths and frequencies. Dispersion relation and damping for the new equations agree better with experimental data than those for the Navier--Stokes--Fourier equations, or the original 13 moments system. The new equations also allow the description of Knudsen boundary layers. [ABSTRACT FROM AUTHOR]

• Devising a class of Fokker-Planck equations consistent with H-theorem. • Devising a kinetic model for arbitrary molecular potential. • Presenting efficient particle solution algorithm for rarefied gas flows. • Testing the devised solution algorithm in high Mach/low speed flows. The diffusion limit of kinetic systems has been subject of numerous studies since prominent works of Lebowitz et al. [1] and van Kampen [2]. More recently, the topic has seen a fresh interest from the rarefied gas simulation perspective. In particular, Fokker-Planck based kinetic models provide novel approximations of the Boltzmann equation, where the relaxation induced by binary collisions is modeled via continuous stochastic processes. Hence in contrast to direct simulation Monte-Carlo, computational particles follow seemingly independent stochastic paths. As a result, a significant computational gain at small/vanishing Knudsen numbers can be obtained, where the dynamics of particles is overwhelmed by collisions. The cubic Fokker-Planck equation derived by [3] gives rise to the correct viscosity and Prandtl number for monatomic gases in the hydrodynamic limit, and further accurate behavior at moderate Knudsen numbers. Yet the model lacks a rigorous structure and more crucially does not admit the H-theorem. The latter underpins its accuracy e.g. in predicting shock wave profiles. This study addresses bridging the gap between diffusion processes and the Boltzmann equation by introducing the Entropic-Fokker-Planck kinetic model. The drift-diffusion closures derived for the model, allow for an H-theorem besides honoring consistent relaxation of moments. The devised model is validated with respect to direct simulation Monte-Carlo for high-Mach as well as Couette flows. Good performance of the model together with its easy to compute coefficients, makes the Entropic-Fokker-Planck framework attractive for computational investigation of gases beyond equilibrium. [ABSTRACT FROM AUTHOR]

The article discusses various reports published within the issue including one by X. J. Gu and D. Emerson investigating boundary value problems for regularized moment equations, another by C. P. T. Groth and J. G. McDonald discussing computational aspects of moment equations, and one by R. S. Myong investigating special variants of the moment method.

• Devising Bayesian inference for fast evaluation of maximum entropy distribution. • Adopting Radial basis function for covariance kernel. • Excellent recovery of bi-modal densities among others. Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations. [ABSTRACT FROM AUTHOR]

• A novel closure named "Entropic Quadrature" (EQ) for moment transport equations of kinetic equations is proposed. • EQ can be interpreted as an extension of the QMOM or an efficient version of the maximum-entropy (ME) closure. • A solver for the Boltzmann equation with configurable order of closure, spatial, and temporal accuracy is described. • Several standard equilibrium and non-equilibrium test cases in one and two dimensions using EQ are conducted. • In the conducted experiments, EQ achieves accuracy comparable to the ME closure at a considerably lower computational cost. Using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the Boltzmann equation to model rarefied gases. One attractive example is the maximum-entropy closure, which, however, is computationally barely affordable. In this paper, we will combine the maximum-entropy approach with the quadrature method of moments (QMOM) – two methods which have in some sense diametric properties, thereby introducing the "Entropic Quadrature" (EQ) closure. EQ calculates a sparse quadrature-based reconstruction of the unknown velocity distribution like the QMOM where the physical meaningful maximum-entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments. We will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions. [ABSTRACT FROM AUTHOR]

In this work, we make a further step in bringing together different approaches to non-equilibrium thermodynamics. The structure of the moment hierarchy derived from the Boltzmann equation is at the heart of rational extended thermodynamics (RET, developed by Ingo Müller and Tommaso Ruggeri). Whereas the full moment hierarchy has the structure expressed in the general equation for the nonequilibrium reversible--irreversible coupling (GENERIC), the Poisson bracket structure of reversible dynamics postulated in that approach is a major obstacle for truncating moment hierarchies, which seems to work only in exceptional cases (most importantly, for the five moments associated with conservation laws). The practical importance of truncated moment hierarchies in rarefied gas dynamics and microfluidics motivates us to develop a new strategy for establishing the full GENERIC structure of truncated moment equations, based on non-entropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of 10 moments. [ABSTRACT FROM AUTHOR]