Numerical Solution Of Kinetic Equation Research Articles

Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

Peculiarities of charged particle kinetics in spherical plasma

We describe transient kinetic simulations of partially ionized weakly-collisional plasma around spherical bodies absorbing or emitting charged particles. Numerical solutions of kinetic equations for electrons and ions in 1D2V phase space are coupled to an electrostatic solver using the Poisson equation or quasineutrality condition for small Debye lengths. The formation of particle groups and their contributions to electric current flow and screening of charged bodies by plasma are discussed for applications to Langmuir probes and solar wind.

Redistribution of the Rydberg State Population Induced by Continuous-Spectrum Radiation

We consider the redistribution of the Rydberg state population resulting from multistep cascade transitions induced by radiation with a continuous spectrum. The population distribution is analyzed within the space of quantum numbers n and l. The dynamics of the system are studied using both the numerical solution of kinetic equations and the diffusion approximation based on the Fokker–Planck equation. The main path of the redistribution process is determined.

Numerical solution of the Boltzmann equation with S-model collision integral using tensor decompositions

The paper presents a new solver for the numerical solution of the Boltzmann kinetic equation with the Shakhov model collision integral (S-model) for arbitrary spatial domains. The numerical method utilizes the Tucker decomposition, which reduces the required computer memory for up to 100 times, even on a moderate velocity grid. This improvement is achieved by representing the distribution function values on a structured velocity grid as a 3D tensor in the Tucker format. The resulting numerical method makes it possible to solve complex 3D problems on modern desktop computers. Our implementation may serve as a prototype code for researchers concerned with the numerical solution of kinetic equations in 3D domains using a discrete velocity method. Program summaryProgram Title: Boltzmann-TCPC Library link to program files:https://doi.org/10.17632/29wv8nmbgn.1Developer’s repository link:https://github.com/chikitkin/Boltzmann-TuckerLicensing provisions: MITProgramming language: Python 3External libraries: Solver is based on the customized version of the tucker3d library [1]Nature of problem: Numerical solution of the Boltzmann kinetic equation with the S-model collision integral in an arbitrary 3D spatial domainSolution method: Discrete velocity method utilizing tensor decomposition for memory reductionAdditional comments including restrictions and unusual features: At present, 1st order advection scheme is used, solver supports unstructured hexagonal meshes written in StarCD ASCII formatReference tucker3d (https://github.com/rakhuba/tucker3d) library contains Python implementations of several important procedures for working with tensors in the Tucker format.

Numerical study of bulk condensation in laminar flow diffusion chamber

The process of bulk condensation was studied on a basis numerical solution of kinetic equation for the mass distribution function of droplet size and the equations of mass and energy balance. The effect of the condenser and preheater deference temperature was studied. Obtained results were compared with other authors’ experimental and numerical data qualitatively and quantitatively.

Limits of the M1 and M2 angular moments models for kinetic plasma physics studies

Angular moments closures are widely used in numerical solutions of kinetic equations. While in the strongly collisional limit they provide a good approximation of the full kinetic equation, their validity domain in the weakly collisional limit is unknown. This work is devoted to defining the validity domain of the M1 model and its extensions, the two populations M1 and the M2 angular moments models for the collisionless kinetic physics applications. Three typical kinetic plasma effects are considered, which are the charged particle beams interaction, the Landau damping and the electromagnetic wave absorption in an overdense semi-infinite plasma. For each case, a perturbative analysis is performed and the dispersion relation is established using the moments models. These relations are compared with those computed by considering the Vlasov equation. The validity limits of each model are demonstrated.

Crystal nucleation kinetics of polyethylene on active centers

Model of nucleation on active centers is applied to formation of folded chain crystals of polyethylene at low supercooling. Nucleation rate and the total number of nuclei are determined by numerical solution of kinetic equations. Thermodynamic and kinetic parameters were chosen to get the coincidence with experimental data for the total number of nuclei and the size distribution of nuclei as a function of time. Nucleation rates for various cluster sizes and the exhaustion of active centers due to the phase transition process were determined. It is shown that the number of exhausted active centers, on which crystalline nuclei are formed, increases with time.

Laser-assisted fragmentation of Al particles suspended in liquid

Laser-assisted fragmentation of Al micro- and nano-particles in liquid is studied both experimentally and theoretically. Modeling of the evolution of size distribution function of particles is performed on the basis of numerical solution of kinetic equation for distribution function which takes into account the ablation and splitting of particles into fragments of different sizes. It is assumed that depending on the size of particles, the fragmentation scenario may vary from monoatomic evaporation to detachment of relatively large fragments. The first mechanism of fragmentation is typical for nanoparticles while the second one is characteristic for large (micrometer-sized) particles. Laser-assisted fragmentation of Al micro-particles suspended in liquid ethanol to Al nanoparticles is experimentally studied using a Nd:YAG laser with pulse duration of 10ps and repetition rate of 300kHz. The evolution of the size distribution function with laser exposure time is monitored and compared to modeling results.

Ion Drag Force on a Charged Macroparticle in Collisionless Plasma

The problem of calculating the ion drag force acting on a charged macroparticle in collisionless flowing plasma is studied by using an approach based on the direct numerical solution of the Vlasov kinetic equations for plasma components. A uniform plasma flow past a spherical macroparticle is considered. The computations are carried out for different particle sizes and different flow velocities. On the basis of the obtained results the effect of particle size on the ion drag force is analyzed. It is shown that when the particle size is much less than the Debye length in plasma, the ion drag force can be calculated with good accuracy by means of the conventional binary collision approach. A modified version of the binary collision approach is proposed to calculate the ion drag force in the case where the particle size becomes comparable to the Debye length in plasma. It is shown that there is a reasonable agreement between the results obtained using the numerical solution of the kinetic equations and that obtained by the modified binary collision approach.

Realizable high-order finite-volume schemes for quadrature-based moment methods

Dilute gas–particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag and particle–particle collisions. However, direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables. An alternative is to reformulate the problem in terms of the moments of the velocity distribution. Recently, a quadrature-based moment method was derived for approximating solutions to kinetic equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from the moments. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. It has been recently shown [14] that realizability is guaranteed only with the 1st-order finite-volume scheme that has an inherent problem of excessive numerical diffusion. The use of high-order finite-volume schemes may lead to non-realizable moments. In the present work, realizability of the finite-volume schemes in both space and time is discussed for the 1st time. A generalized idea for developing realizable high-order finite-volume schemes for quadrature-based moment methods is presented. These finite-volume schemes give remarkable improvement in the solutions for a certain class of problems. It is also shown that the standard Runge–Kutta time-integration schemes do not guarantee realizability. However, realizability can be guaranteed if strong stability-preserving (SSP) Runge–Kutta schemes are used. Numerical results are presented on both Cartesian and triangular meshes.

Formation of crystal nuclei near critical supersaturation in small volumes

This work deals with the nucleation of crystals in confined systems in response to the recent high interest in research on crystallization in emulsion and microemulsion droplets. In these confined systems, crystallization often occurs at high supercooling; thus, nucleation determines the overall crystallization process. A decrease in the volume of the confined mother phase leads to the higher supercooling needed for the phase transition. We have numerically solved kinetic equations in order to determine the conditions under which the first crystal nuclei are formed by homogeneous and heterogeneous nucleation from supercooled melt and supersaturated solution, depending on the volume of the mother phase. Supersaturation (or supercooling) increases with decreasing volume of the mother phase. The nucleation barrier depends linearly on the logarithm of volume of the mother phase in all cases under consideration, as follows from the numerical solution of kinetic equations.

Mini-Workshop: Numerics for Kinetic Equations

The workshop Numerics for Kinetic Equations , organized by Irene Gamba (Austin, USA), Sergej Rjasanow (Saarbrücken, Germany), and Wolfgang Wagner (Berlin, Germany) was held November 16th–November 22nd, 2008. The meeting was well attended with 15 participants (11 from Europe, 3 from USA, and 1 from Japan). This workshop brought together researchers with various backgrounds: mathematics, physics, chemistry, and engineering science. The interdisciplinarity of the invited participants was essential because the numerical solution of kinetic equations is characterized by a wide area of applications (rarefied gas dynamics, modeling of semiconductors, plasma, or combustion). Furthermore, the strong coupling between different areas of science is needed to understand the complicated and highly nonlinear nature of kinetic equations. The last decades have shown an amazing development in the design of effective numerical methods for kinetic equations describing processes in micro- and nanomachines. A particular example is the Knudsen compressor – an energy efficient micro-scale pump with no moving parts. Furthermore, the inelastic Boltzmann equation is more and more used in applications. Finally, semiconductor transport models are still a challenge for numerics. In this mini-workshop, we have discussed recent mathematical and computational issues, mainly related to the so called “low Mach number” flows, from the point of view of numerical efficiency and accuracy. When solving those problems by the use of the Direct Simulation Monte Carlo method, one has to deal with a small signal-to-noise ratio. Thus, due to statistical fluctuations, it is difficult to compute the physical flow parameters (density, stream velocity, temperature, and, especially, heat flux) with sufficient accuracy, so variance reduction methods are desperately needed. On the other hand, deterministic algorithms do not exhibit statistical fluctuations, and, therefore, may become competitive for slow flows. The talks have been devoted to Furthermore, three extended “round table” discussions were devoted to These discussions were extremely useful for facilitating a clear understanding of the main difficulties when applying stochastic and/or deterministic methods to kinetic equations close to the hydrodynamic limit. The participants appreciated the idea of the mini-workshops – to have not too many talks, but instead plenty of time for intensive and detailed discussions. We would strongly support further such workshops. Finally, we would like to thank both the administration and the staff of the MFO for providing excellent working conditions and creating a pleasant atmosphere.

Homogeneous nucleation rate at various initial supersaturations in a closed system

The rate of formation of nuclei in a closed system is studied by numerical solution of kinetic equations at various supersaturations. Depletion of vapor (decrease of supersaturation) due to the phase transition process is taken into account. At a relatively low value of initial supersaturations, the decrease of supersaturation is negligible. At somewhat higher initial supersaturations, the phase transition process is more intense and the decrease of supersaturation plays an important role. As a consequence the nucleation rate after reaching a maximum decreases with time and for sufficiently long times only the nuclei of certain size are formed. At high initial supersaturation the nucleation rate for subcritical cluster sizes is negative, i.e. the number of subcritical clusters tends to get reduced. Critical size increases with time due to lowered supersaturation.

New method of simulation of volume condensation of supersaturated vapor

A method of direct numerical solution of kinetic equation for the droplet size distribution function is suggested for the investigation of the process of volume condensation of supersaturated vapor. Use is made of analogy with the corresponding method of solution of kinetic Boltzmann equation. An advantage of the suggested approach is the absence of limitations on the Knudsen number. The method is tested using the example of simulation of the behavior of vapor under conditions of rapid development of the state of supersaturation in a vapor-gas mixture as a result of adiabatic expansion. The results are compared in a wide range of Knudsen number with the results obtained using the moments method in a version of a set of moments equations. Investigation is performed of the effect of the droplet size dependence of saturation pressure on the dynamics of the process of condensation relaxation of supersaturated vapor.

Temporal and Spatial Organization of Chemical and Hydrodynamic Processes. The System Pb2+−Chlorite−Thiourea

Precise spatio-temporal organization of chemical, hydrodynamic, and mechanical processes is typical for biological systems where particular chemical reactions have to accrue in precisely assignment place and time. It is rarely studied and observed in chemical systems. We report unusual precipitation pattern formation of PbSO(4) in chemical media (Pb(2+)-Chlorite-Thiourea System). We have found that there is a region in a plane of initial concentrations of chlorite ions and thiourea where precipitation of lead sulfate appears in a form of ring if a pellet of lead nitrate is placed into the system. The whole process may be divided into three stages: movement of first circular front of lead containing solution, formation of a ringlike pattern of lead sulfate, and finally, propagation of this pattern resulting in a formation of ring with final inside diameter. Our experiments indicate that the following values are reproducible and quantify the PbSO(4) ring evolution: induction time, radius of the ring birth, speed of ring propagation toward the center, and final inside radius of the ring. Numerical solution of kinetic equations allowed us to give a qualitative explanation for the phenomenon observed. Formation and evolution of the PbSO(4) rings are caused by interplay of concentration gradients in the system and chemical reactions that occur in excitable chlorite-thiourea system. Chemical reactions and hydrodynamic processes form a complex causal network that made morphogenesis of this unusual pattern possible.

A multiscale model for kinetics of formation and disintegration of spherical micelles

Dynamics of self-assembly and structural transitions in surfactant systems often involve a large span of length and time scales. A comprehensive understanding of these processes requires development of models connecting phenomena taking place on different scales. In this paper, we develop a multiscale model for formation and disintegration of spherical nonionic micelles. The study is performed under the assumption that the dominant mechanism of micelle formation (disintegration) is a stepwise addition (removal) of single monomers to (from) a surfactant aggregate. Different scales of these processes are investigated using a combination of coarse-grained molecular dynamics simulations, analytical and numerical solution of stochastic differential equations, and a numerical solution of kinetic equations. The removal of a surfactant from an aggregate is modeled by a Langevin equation for a single reaction coordinate, the distance between the centers of mass of the surfactant and the aggregate, with parameters obtained from a series of constrained molecular dynamics simulations. We demonstrate that the reverse process of addition of a surfactant molecule to an aggregate involves at least two additional degrees of freedom, orientation of the surfactant molecule and micellar microstructure. These additional degrees of freedom play an active role in the monomer addition process and neglecting their contribution leads to qualitative discrepancies in predicted surfactant addition rates. We propose a stochastic model for the monomer addition which takes the two additional degrees of freedom into account and extracts the model parameters from molecular dynamics simulations. The surfactant addition rates are determined from Brownian dynamics simulations of this model. The obtained addition and removal rates are then incorporated into the kinetic model of micellar formation and disintegration.

Study of the volume condensation process in supersaturated vapor by the direct numerical solution of the kinetic equation for the droplet size distribution function

The volume condensation of supersaturated vapor is investigated by the direct numerical solution of the basic kinetic equation for the droplet size distribution function by analogy with the corresponding solution of the Boltzmann kinetic equation. The proposed consideration of the condensation growth of droplets is applicable at any Knudsen number. The method is tested by the example of vapor condensation under the conditions of the rapid development of supersaturation in a vapor-gas mixture as a result of its adiabatic expansion. In a wide range of Knudsen numbers, the results of the modeling are compared with those obtained by the moment method.

Cluster growth and dissolution of fullerenes in non-polar solvents

The processes of fullerene cluster formation and growth in non-polar solvents are described basing on the numerical solutions of kinetic equations of nucleation theory. The liquid drop model and the limited growth model are analyzed. For supersaturated solutions it is shown, that the model of limited growth is an adequate model and can qualitatively describe the experimental data. The model of fullerene dissolution processes is proposed.

Formation of droplets on active centers in supersaturated vapors

Kinetics of nucleation on active centers in supersaturated vapors is studied by a numerical solution of kinetic equations, including depletion of active centers during the phase transition process. The basic characteristics of the nucleation process (the size distribution of nuclei, nucleation rate and the total number of nuclei) are determined at various conditions. Nuclei are formed on all active centers at a sufficiently long time in coincidence with a widely used Avrami model. An increase of the total number of nuclei with time is steeper in the Avrami model as compared with our numerical solution of kinetic equations. The nucleation rate does not reach its stationary limit and becomes negligibly small with increasing time.

Numerical consideration of quasilinear electron cloud dynamics in plasma

The dynamics of a hot electron cloud in the solar corona-like plasma based on the numerical solution of kinetic equations of weak turbulence theory is considered. Different finite difference schemes are examined to fit the exact analytical solutions of quasilinear equations in hydrodynamic limit (gas-dynamic solution). It is shown that the scheme suggested demonstrates correct asymptotic behavior and can be employed to solve initial value problems for an arbitrary initial electron distribution function.