Initial Distribution Function Research Articles

Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems

This article introduces a refined mathematical model to evaluate the quality of mobile radio channels within cyber-physical systems, employing the topological transformation of stochastic networks. The operation of the radio channel is conceptualized as a stochastic network, enabling the derivation of critical metrics such as an equivalent function, mathematical expectation, variance, and the time distribution function of the implemented processes. The model uses the Gamma distribution for the initial distribution functions of random variables, enhancing its analytical precision. A significant advancement of this study is the development of a comprehensive model that describes the data transmission process through phases of connection establishment, information transmission, and connection maintenance. The innovative aspect of this research lies in applying an equivalent function to a stochastic network that includes a logical “AND” node with gamma-distributed incoming branches. The stochastic network presented in the article, which includes a logical “AND” node, helps to elucidate the mechanism for obtaining an equivalent function for such networks, allowing the application area of the GERT method to be expanded. This methodological enhancement extends the previously limited scope of topological transformation methods, which only applied to exponential distribution models for the timing of branch inputs. By integrating a Gamma distribution, the model simplifies the equivalent function and reduces the computational complexity required to assess the radio channel’s quality, ensuring the accuracy needed for engineering calculations. Moreover, the proposed method requires 25–40% fewer series members than the traditional Taylor series decomposition, while maintaining comparable computational complexity for the typical series members. Furthermore, the maximum absolute error in the calculations is capped at 9 × 10−3, which is well within acceptable limits for engineering purposes. Primarily designed for radio channels in cyber-physical systems, the model’s applicability extends to wireless communications, providing a valuable tool for evaluating channel efficiency and security in the face of increasing cyber threats.

Wave Generation by Flare-accelerated Ions and Implications for 3He Acceleration

The waves generated by high-energy proton and alpha particles streaming from solar flares into regions of colder plasma are explored using particle-in-cell simulations. Initial distribution functions for the protons and alphas consist of two populations: an energetic, streaming population represented by an anisotropic (T ∥ > T ⊥), one-sided kappa function and a cold, Maxwellian background population. The anisotropies and nonzero heat fluxes of these distributions destabilize oblique waves with a range of frequencies below the proton cyclotron frequency. These waves scatter particles out of the tails of the initial distributions along constant-energy surfaces in the wave frame. Overlap of the nonlinear resonance widths allows particles to scatter into near-isotropic distributions by the end of the simulations. The dynamics of 3He are explored using test particles. Their temperatures can increase by a factor of nearly 20. Propagation of such waves into regions above and below the flare site can lead to heating and transport of 3He into the flare acceleration region. The amount of heated 3He that will be driven into the flare site is proportional to the wave energy. Using values from our simulations, we show that the abundance of 3He driven into the acceleration region should approach that of 4He in the corona. Therefore, waves driven by energetic ions produced in flares are a strong candidate to drive the enhancements of 3He observed in impulsive flares.

On the Existence of Global Solutions for the Vlasov–Poisson System in a Half-space and Plasma Confinement

We consider the Vlasov–Poisson system with external magnetic field in a half-space with the Neumann boundary condition for the electric potential and specular reflection on a boundary. For arbitrary compactly supported initial density distribution functions, we obtain sufficient conditions for external magnetic field, which provide global existence of density distribution functions with compact supports lying at some distance from a boundary.

VLF chorus emissions modeling using EPOCH PIC code: Generation regimes and comparison with a backward wave oscillator theory

We present the results of a numerical model for VLF chorus emissions generation based on the EPOCH PIC code. The initial distribution function of energetic elections is assumed to have a finite-width step-like deformation in the velocities parallel to the geomagnetic field. This allows us to verify and generalize the analytical and numerical results obtained earlier within the framework of a backward wave oscillator model in which a sharp step feature is assumed. We show that in the presence of a sharp or smooth step, chorus emissions can be generated for realistic flux of energetic electrons even if the initial velocity distribution (without the step) is isotropic. Simulated chorus emissions behave very similarly for both sharp and finite-width steps. In particular, the generation of repetitive chirping chorus elements occurs if the coupling parameter exceeds the linear generation threshold several times. For smaller threshold excess, quasi-stationary or relaxation oscillation regimes take place. For the first time, we systematically study the dependence of the chorus emissions properties on the energetic electron distribution parameters, in particular, on the step width and the effective wave-particle coupling parameter proportional to the energetic electron flux and step height. We show that the functional dependences of the linear growth rate on the coupling parameter is the same for a smooth step gradient as those obtained for the sharp step within the framework of the backward wave oscillator model. The threshold flux of energetic electrons, above which whistler mode waves are generated, increases very fast with increasing step width. With a fixed relative step height, the threshold flux is about an order of magnitude higher for a step width of 8% than for a sharp step.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

Predicting pesticide residues in pod fruits with a modified peel-like uptake model: A green pea demonstration

Peas are among the most popular leguminous plants, consumed by both humans and animals in large quantities. Pesticides are widely used globally to increase pea yield, and as a result, pesticide residues can be taken up by pea plants and bioaccumulate in their fruits, including peas and pods. However, there is a lack of modeling approaches available to predict residue concentrations in peas. To address this issue, a pod fruit model (specifically designed for neutral organic compounds) was proposed to simulate the bioaccumulation process of pesticide residues in pea plants, which was developed by modifying a peel-like uptake model. The simulation results, based on green pea as the modeling demonstration, reveal that moderately-lipophilic pesticides (i.e., log KOW around 3) have higher simulated concentrations in peas at harvest compared to hydrophilic (i.e., log KOW less than 0) or highly-lipophilic (i.e., log KOW over 5) pesticides, which is due to the enhanced uptake process of moderately-lipophilic compounds in the pod-pea system, such as their ability to penetrate the pod cuticle and be transported via phloem sap. The sensitivity test and variability analysis conducted in this study revealed that the degradation kinetics, including metabolism, hydrolysis, and photolysis, had a significant impact on moderately-lipophilic pesticides due to their high simulated concentrations in the pea plant. This can result in substantial loss of residue mass via degradation. The validation of the model demonstrated that the simulation results, specifically residue concentrations in the fruit, were consistent with the harvested data. However, some inconsistency was observed immediately after pesticide application, which could be attributed to plant growth dynamics and initial surface mass distributions. The proposed pod fruit model provides new insights into the bioaccumulation process of pesticide residues in pea plants and enables high-throughput simulations of residue concentrations at harvest. To enhance the performance of the pod fruit model, future research should consider plant growth dynamics, plant uptake of ionizable compounds, and initial mass distribution functions.

Detailed analysis of banded chorus gap formation by an electron shell distribution

In a recent paper [K. Min, Phys. Plasmas 30, 012904 (2023)], the formation of the banded chorus with a gap in intensity at half the electron cyclotron frequency (Ωe/2) is demonstrated by particle-in-cell simulations including an isotropic shell distribution at an intermediate energy. This follow-up study focuses on the phase space density (PSD) hill formation process and its role in the chorus wave damping at the gap. We first show that phase-trapped particles closely follow single wave characteristics in momentum space. This means that the formation of either PSD hole or hill is primarily determined by the temperature anisotropy, T⊥/T‖, of an initial distribution function. The critical value of T⊥/T‖ increases (decreases) for a higher (lower) resonant frequency. We then revisit the recent banded chorus simulations to investigate how the presence of an isotropic shell distribution self-consistently affects chorus wave evolution at the gap. Initially, with an increasing wave frequency, more and more shell electrons get trapped and a PSD hill is formed. The enhanced PSD hill counteracts wave growth driven by phase-trapped anisotropic electrons and subsequently reduces wave amplitude. The weakened wave self-consistently feeds back to the particle trapping, ultimately suppressing both the PSD hole and hill. By the time the wave frequency reaches about 0.45Ωe, the gyro-phase structure of the electron distribution becomes much less organized. In some cases, however, the wave growth at the upstream source region can be strong enough that waves still manage to go through the gap frequency, suggesting that additional process(es) should likely be accompanied.

The energy distribution of the first supernovae

ABSTRACT The nature of the first Pop III stars is still a mystery and the energy distribution of the first supernovae is completely unexplored. For the first time we account simultaneously for the unknown initial mass function (IMF), stellar mixing, and energy distribution function (EDF) of Pop III stars in the context of a cosmological model for the formation of a MW-analogue. Our data-calibrated semi-analytic model is based on a N-body simulation and follows the formation and evolution of both Pop III and Pop II/I stars in their proper time-scales. We discover degeneracies between the adopted Pop III unknowns, in the predicted metallicity and carbonicity distribution functions and the fraction of C-enhanced stars. None the less, we are able to provide the first available constraints on the EDF, $dN/dE_\star \propto E_{\star }^{-\alpha _e}$ with 1 ≤ αe ≤ 2.5. In addition, the characteristic mass of the Pop III IMF should be mch < 100 M⊙, assuming a mass range consistent with hydrodynamical simulations (0.1–1000 M⊙). Independent of the assumed Pop III properties, we find that all $\rm [C/Fe]\gt +0.7$ stars (with $\rm [Fe/H]\lt -2.8$) have been enriched by Pop III supernovae at a $\gt 20~{{\ \rm per\ cent}}$ level, and all $\rm [C/Fe]\gt +2$ stars at a $\gt 95~{{\ \rm per\ cent}}$ level. All very metal-poor stars with $\rm [C/Fe]\lt 0$ are predicted to be predominantly enriched by Pop III hypernovae and/or pair instability supernovae. To better constrain the primordial EDF, it is absolutely crucial to have a complete and accurate determination of the metallicity distribution function, and the properties of C-enhanced metal-poor stars (frequency and [C/Fe]) in the Galactic halo.

Electrostatic upper-hybrid mode instability driven by a ring electron distribution

Quasi electrostatic fluctuations in the upper-hybrid frequency range are commonly detected in the planetary magnetospheric environment. The origin of such phenomena may relate to the instability driven by a loss-cone feature associated with the electrons populating the dipole-like magnetic field. The present paper carries out a one-dimensional electrostatic particle-in-cell simulation accompanied by a reduced quasilinear kinetic theoretical analysis to investigate the dynamics of the upper-hybrid mode instability driven by an initial ring electron distribution function, which is a form of loss-cone distribution. A favorable comparison is found between the two approaches, which shows that the reduced quasilinear theory, which is grounded in the concept of a model of the particle distribution function that is assumed to maintain a fixed mathematical form except that the macroscopic parameters that define the distribution are allowed to evolve in time, can be an effective tool in the study of plasma instabilities, especially if it is guided by and validated against the more rigorous simulation result.

Truncated Moments for Heavy-Tailed and Related Distribution Classes

Suppose that ξ+ is the positive part of a random variable defined on the probability space (Ω,F,P) with the distribution function Fξ. When the moment Eξ+p of order p>0 is finite, then the truncated moment F¯ξ,p(x)=min1,Eξp1I{ξ>x}, defined for all x⩾0, is the survival function or, in other words, the distribution tail of the distribution function Fξ,p. In this paper, we examine which regularity properties transfer from the distribution function Fξ to the distribution function Fξ,p and which properties transfer from the function Fξ,p to the function Fξ. The construction of the distribution function Fξ,p describes the truncated moment transformation of the initial distribution function Fξ. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that Fξ,p belongs to one of these classes, it follows that Fξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.

Adaptive partitioning-based discrete unified gas kinetic scheme for flows in all flow regimes

To improve the efficiency of the discrete unified gas kinetic scheme (DUGKS) in capturing cross-scale flow physics, an adaptive partitioning-based discrete unified gas kinetic scheme (ADUGKS) is developed in this work. The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation, which contains the initial distribution function and the local equilibrium state. The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes. When the contribution of the initial distribution function is negative, the local flow field can be regarded as the continuous flow and the Navier–Stokes (N-S) equations can be used to obtain the solution directly. Otherwise, the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefaction effect. Given this, in the ADUGKS, the computational domain is divided into the DUGKS cell and the N-S cell based on the contribution of the initial distribution function to the calculation of free streaming fluxes. In the N-S cell, the local flow field is evolved by solving the N-S equations, while in the DUGKS cell, both the discrete velocity Boltzmann equation and the corresponding macroscopic governing equations are solved by a modified DUGKS. Since more and more cells turn into the N-S cell with the decrease of the Knudsen number, a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.

Effect of the thermalization of secondary electron energies on plasma formation during the irradiation of heavy ions

This study aims at understanding the effect of the thermalization of secondary electrons produced by incident heavy-ion impact ionization on plasma formation. The motions of the secondary electrons were simulated using pseudo-initial secondary electron energy distribution functions. These simulations allowed us to estimate the times when thermalization occurred and the probabilities of secondary electrons escaping the electric field created near the incident ion path according to the initial secondary electron energy distribution functions. The probabilities and thermalization processes were found to be almost independent of these functions which indicated that the secondary electrons were thermalized before the plasma was formed.

Homogeneous generalized master equation for an arbitrary initial state of a many-particle classical system

A special projection operator is introduced, which exactly transforms the Nakajima–Zwanzig inhomogeneous Generalized Master Equation (GME) with an irrelevant initial condition term (a source) into a completely closed homogeneous GME (with no initial correlations term) for the relevant part of an N-particle distribution function FN(t) for an arbitrary initial distribution function FN(0) of a system of classical particles. It resolves the problem of the derivation of a closed linear evolution equation from the Liouville equation with no “molecular chaos” type approximation. The obtained equation is equivalent to the completely closed equation for s-particle (s≤N) distribution function Fs(t). The initial correlations are accounted for in this equation in the modified by the introduced projection operator kernel governing the evolution of Fs(t). This equation is further simplified by presenting its kernel in the linear in the particles’ density n approximation. In this approximation the equations for one-particle F1(t) and two-particle F2(t) distribution functions are derived. Special attention is given to analyzing the equation for F1(t) and showed that this equation in the Markov approximation can be written as the linear Boltzmann or Landau (for a small interparticle interaction) equations but with additional terms caused by initial correlations.

Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlation at the start of the simulation between the statistical fluctuations of the weighted densities of macroparticle centers in the plasma particle phase-space. The fluctuations are associated with different initial conditions, typically due to the random initial conditions (in velocity space) of the macroparticles/simulation plasma, assigned according to their initial distribution of probability. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace–Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell’s equations for the fields, as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelation spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon (1970b Proc. 4th Conf. Numerical Simulation of Plasmas) using a test macroparticle approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes using arbitrary macroparticle weights. Analytical estimates of statistical fluctuation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles. This method is illustrated by linear Landau damping uncovering (from noise, which is usually considered numerical) the physical fluctuations driven by a single small amplitude electrostatic wave perturbing a PIC simulation plasma in equilibrium.

Two-component self-gravitating isothermal slab models

We revisit the problem of the isothermal slab (in standard Cartesian coordinates, density distributions and mean gravitational potential are considered to be independent of x and y and to be a function of z, symmetric with respect to the {{z}} = {{0}} plane) in the context of the general issues related to the role of weak collisionality in inhomogeneous self-gravitating stellar systems. We thus consider the two-component case, that is a system of heavy and light stars with assigned mass ratio ({varvec{mu }}) and assigned global relative abundance ({{alpha }}; the ratio of the total mass of the heavy and light stars). The system is imagined to start from an initial condition in which the two species are well mixed and have identical spatial and velocity distributions and to evolve into a final configuration in which collisions have generated equipartition and mass segregation. Initial and final distribution functions are assumed to be Maxwellian. Application of mass and energy conservation allows us to derive the properties of the final state from the assumed initial conditions. In general, the derivation of these properties requires a simple numerical integration of the Poisson equation. Curiously, the case in which the heavy stars are exactly twice as massive as the light stars ({{mu }} = {{2}}) turns out to admit a relatively simple analytic solution. Although the general framework of this investigation is relatively straightforward, some non-trivial issues related to energy conservation and the possible use of a virial constraint are noted and clarified. The formulation and the results of this paper prepare the way to future studies in which the evolution induced by weak collisionality will be followed either by considering the action of standard collision operators or by means of dedicated numerical simulations.

Development of explicit formulations of G45-based gas kinetic scheme for simulation of continuum and rarefied flows.

In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.

On the Phase-mixed Eccentricity and Inclination Distributions of Wide Binaries in the Galaxy

Modern observational surveys allow us to probe the distribution function (DF) of the Keplerian orbital elements of wide binaries in the solar neighborhood. This DF exhibits nontrivial features, in particular a superthermal distribution of eccentricities for semimajor axes a ≳ 103 au. To interpret such features we must first understand how the binary DF is affected by dynamical perturbations, which typically fall into two classes: (i) stochastic kicks from passing stars, molecular clouds, etc. and (ii) secular torques from the Galactic tide. Here we isolate effect (ii) and calculate the time-asymptotic, phase-mixed DF for an ensemble of wide binaries under quadrupole-order tides. For binaries wide enough that the phase-mixing assumption is valid, none of our results depend explicitly on semimajor axes, masses, etc. We show that unless the initial DF is both isotropic in binary orientation and thermal in eccentricity, then the final phase-mixed DF is always both anisotropic and nonthermal. However, the only way to produce a superthermal DF under phase mixing is for the initial DF to itself be superthermal.

Efficient subset simulation for rare-event integrating point-evolution kernel density and adaptive polynomial chaos kriging

Rare-event probability estimation has a wide range of applications, including the design and manufacture of precision equipment, aerospace systems, and critical industrial and civil structures. However, traditional simulation-based reliability calculation methods, such as brute Monte Carlo simulation (MCS) and subset simulation (SS), face challenges in efficiently evaluating small-failure probabilities due to the need for a large number of simulations, especially for non-linear and complex scenarios. Thus, to efficiently assess the probability of rare failure events in structural engineering, this paper develops a novel method for assessing the small-failure probability by integrating the point-evolution kernel density (PKDE), SS, and polynomial chaos kriging (PCK). The proposed PKDE-Adaptive PCK-based SS (PAPS) method aims to reduce the implementation of the original performance function by PCK and enrich the training set using an adaptive strategy. Moreover, the initial cumulative distribution function (CDF) of the performance function estimated by PKDE is modified gradually to facilitate the estimation of small-failure probability. Four numerical examples of small-failure probability estimation involving classical analytical cases, time-variant cases, and non-linear stochastic structures are used to illustrate the accuracy and efficiency of the proposed method. The computational results show that the proposed method can provide accurate computational results with a smaller computational burden than traditional methods (e.g., MCS, SS, LHS-PCK-SS).

Grad's distribution functions-based gas kinetic scheme for simulation of flows beyond Navier–Stokes level

Conventional gas kinetic scheme (GKS) has been successfully utilized to obtain the accurate solution of Navier–Stokes (NS) equations. However, when it comes to flows beyond NS level, the conventional GKS is not reliable because its initial gas distribution function is approximated by the first-order Chapman–Enskog expansion, which merely recovers the NS equations. In order to make an extension for flows beyond NS level, we propose the Grad's distribution functions-based GKS in this paper. This scheme retains the advantage of conventional GKS and constructs the numerical fluxes through a time-dependent gas distribution function, which is derived from the integral solution of Boltzmann equation. In the present scheme, the initial gas distribution function in the local solution of Boltzmann equation is approximated by Grad's 13 and 26 distribution functions. Furthermore, the high-order moments in the initial Grad's distribution function are calculated by moment relationship directly, and thus, the solution of complicated partial differential equations for these high-order moments is avoided. Four benchmark numerical examples are tested to validate the performance of the present scheme, and the results demonstrate that the present scheme can not only recover NS solutions in the continuum region but also predict reasonable results for flows in the slip and transition regimes.

Variant of gas kinetic flux solver for flows beyond Navier-Stokes level.

In this paper, a variant of gas kinetic flux solver (GKFS) is presented for simulation of flows beyond the Navier-Stokes (NS) level. The method retains the framework of GKFS and reconstructs the numerical fluxes by the moments of distribution function at the cell interface, which is given from the local solution of the Boltzmann equation. In the conventional GKFS, the first-order Chapman-Enskog (CE) expansion is utilized to approximate the initial distribution function. By using the differential chain rule, it was found that the CE expansion form could be linked to the stress tensor and the heat flux. For flows in the NS level, the stress tensor and heat flux can be simply calculated from the linearized constitutive relationship and Fourier's law, respectively. However, for flows beyond the NS level, due to the strong nonequilibrium effect, the linearized constitutive relationship and Fourier's law are insufficient to predict the stress tensor and the heat flux. To overcome this difficulty, this paper introduces correction terms to the stress tensor and heat flux in the initial distribution function. These correction terms will take effect in the strong nonequilibrium region for flows beyond the NS level. To avoid finding complex expressions or solving complicated partial differential equations for the correction terms, a simple and iterative procedure is proposed to update the correction terms based on the framework of GKFS. The proposed method is validated by three benchmark cases which cover the flow from the continuum regime to the transition regime. Numerical results show that the present solver can provide accurate solution in the continuum regime. It is indeed the correction terms that take effect in the strong nonequilibrium region for flows beyond the NS level, which enables the present solver to capture the nonequilibrium phenomenon with reasonable accuracy for rarefied flows at moderate Knudsen number.