Chapman-Enskog Expansion and Transport Coefficients for the NJL Model

The transport equation for the Nambu-Jona-Lasinio (NJL) model isderived phenomenologically. The finite-temperature effective mass for the quarkis analysed, By means of the Chapman-Enskog method and hydrodynamic ap-proach the different transport coefficients for the NJL plasma are calculated tothe first order in the relaxation time.

Modified Chapman–Enskog expansion: A new way to treat divergent series

The resolution by Chen and Sun of divergent Chapman-Enskog expansion problem will not only build a unified foundation for non-equilibrium dynamics modeling at all Mach number and Knudsen number, but also shed light to a large class of difficult theoretical problems involving divergent expansion on strong nonlinearity.

Asymptotic Expansions for Solutions of Parabolic Systems Associated with Multi-scale Switching Diffusions

This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.

A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion

In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations （NPDEs） in the form ut ＋αuux ＋βu^nuz ＋γuxx ＋δuzxx ＋ζxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

Numerical Solution of Nonlinear Klein-Gordon Equation Using Lattice Boltzmann Method

In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L2, L∞ and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.

MICROCOSMIC BOUND THEOREM OF DAYCY'S LAW AND ITS APPLICATION

By combining Chapman-Enskog expansion with the BGK approximation to Baltzmann equation and Navier-Stokes equation was obtained. And an expression of (Darcy's) law was obtained through taking variable average over Navier-Stokes equation on some representative space in porous media,and finally an example was taken to prove its reliability.

A higher order lattice BGK model for simulating some nonlinear partial differential equations

In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.

Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method

In this paper,we develop a lattice Boltzmann model for a class ofone-dimensional nonlinear wave equations,including the second-order hyperbolictelegraph equation,the nonlinear Klein-Gordon equation,the damped and undampedsine-Gordon equation and double sine-Gordon equation.By choosing properly theconservation condition between the macroscopic quantity u,and the distributionfunctions and applying the Chapman-Enskog expansion,the governing equation isrecovered correctly from the lattice Boltzmann equation.Moreover,the local equilib-rium distribution function is obtained.The results of numerical examples have beencompared with the analytical solutions to confirm the good accuracy and the applica-bility of our scheme.

Two-Relaxation-Time Lattice Boltzmann Scheme:About Parametrization,Velocity,Pressure and Mixed Boundary Conditions

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamicequations with variable source terms based on equivalent equilibriumfunctions. A special parametrization of the free relaxation parameter is derived. Itcontrols, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopicsteady solution and governs the spatial discretization of transient flows. Inthis framework, the multi-reflection approach [16, 18] is generalized and extended forDirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions.We propose second and third-order accurate boundary schemes and adapt themfor corners. The boundary schemes are analyzed for exactness of the parametrization,uniqueness of their steady solutions, support of staggered invariants and for the effectiveaccuracy in case of time dependent boundary conditions and transient flow.When the boundary scheme obeys the parametrization properly, the derived permeabilityvalues become independent of the selected viscosity for any porous structureand can be computed efficiently. The linear interpolations [5, 46] are improved withrespect to this property.

Heat Transfer Problem for the Boltzmann Equation in a Channel with Diffusive Boundary Condition

In this paper,the authors study the 1 D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.This problem was studied by Esposito-Lebowitz-Marra(1994,1995)where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition.However,a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary.In the regime where the Knudsen number is reasonably small,the slip phenomenon is significant near the boundary.Thus,they revisit this problem by taking into account the slip boundary conditions.Following the lines of[Coron,F.,Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation,J.Stat.Phys.,54(3-4),1989,829-857],the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points.Then they will establish a uniform L∞estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

In this paper,a simplified lattice Boltzmann method(SLBM)without evolution of the distribution function is developed for simulating incompressible viscous flows.This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes(N-S)equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis.In SLBM,the equilibrium distribution function is calculated from the macroscopic variables,while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions.Therefore,SLBM tracks the evolution of the macroscopic variables rather than the distribution function.As a result,lower virtual memories are required and physical boundary conditions could be directly implemented.Through numerical test at high Reynolds number,the method shows very nice performance in numerical stability.An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space.More benchmark tests,including the Couette flow,the Poiseuille flow as well as the 2D lid-driven cavity flow,are conducted to further validate the present method;and the simulation results are in good agreement with available data in literatures.

A Gas Kinetic Scheme for the Simulation of Compressible Multicomponent Flows

In this paper,a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK model in[A.D.Kotelnikov and D.C.Montgomery,A Kinetic Method for Computing Inhomogeneous Fluid Behavior,J.Comput.Phys.134(1997)364-388].Different from the conventional BGK model,the collisions between different species are taken into consideration.Based on the Chapman-Enskog expansion,the corresponding macroscopic equations are derived from this two-species model.Because of the relaxation terms in the governing equations,the method of operator splitting is applied.In the hyperbolic part,the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method.Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows.The theoretical analysis on the spurious oscillations at the interface is also presented.

Elements of the Lattice Boltzmann Method II: Kinetics and Hydrodynamics in One Dimension

Concepts of the lattice Boltzmann method are discussed in detail for the one-dimensional kinetic model.Various techniques of constructing lattice Boltzmann models are discussed,and novel collision integrals are derived.Geometry of the ki-netic space and the role of the thermodynamic projector is elucidated.

Lattice Boltzmann Modeling of Viscous Elementary Flows

A lattice Boltzmann method is developed for modeling viscous elementary flows.An adjustable source term is added to the lattice Boltzmann equation,which can be tuned to model different elementary flow features like a doublet or a point source of any strength,including a negative source(sink).The added source term is dimensionally consistent with the lattice Boltzmann equation.The proposed model has many practical applications,as it can be used in the framework of the potential flow theory of viscous and viscoelastic fluids.The model can be easily extended to the three dimensional case.The model is verified by comparing its results with the analytical solution for some benchmark problems.The results are in good agreement with the analytical solution of the potential flow theory.