many emerging nanostructures involve semiconductors and dielectrics , in which phonon transport is the main mechanism for heat transfer .heat transfer process in systems with such nanostructures usually involves multiple temporal and spatial scales , and it is a challenging problem to develop efficient numerical methods that are applicable to different transport regimes .owing to the breakdown of fourier law at small time and spatial scales , and the high computational requirement of microscopic molecular dynamics , the phonon boltzmann transport equation ( bte ) is regarded to be able to provide a good base for developing numerical methods for multiscale heat transfer when the wave effect of phonon is negligible .actually , many numerical schemes have been proposed to solve the bte in previous studies , including the stochastic monte - carlo ( mc ) method and the deterministic discrete ordinates method ( dom ) coupled with finite - difference , finite - volume , or finite - element discretization of spatial space . the lattice boltzmann method ( lbm ) , which was originally developed for continuous fluid flows , was also applied to phonon transport .  generally , the mc method follows a time - splitting algorithm , namely , the dynamics of a simulated particle is decoupled into advection and scattering processes , and thus the time step used is less than the relaxation time , and the grid size is less than the phonon mean - free - path . consequently , the computational costs of mc method are expensive in the acoustic thick regime , which prohibit its applications for multiscale problems with diffusive region , although it can be quite efficient for ballistic transport .it is also noted that an improved mc method has been developed recently by simulating only the deviation from equilibrium such that the variance can be efficiently reduced in simulating systems with small temperature variations . in the dom method , the transient and advection terms in the bte are usually discretized with techniques that are adopted in computational fluid dynamics ( cfd ) , such as upwind ( step ) and central ( diamond ) finite - difference schemes , or finite - volume schemes with upwind interpolations .these cfd techniques may introduce significant artificial diffusions ( low - order schemes ) or numerical instability ( high - order schemes ) . regarding the lbm for phonon transport , although it has been applied to some nano and multiscale problems , some studies have shown that lbm may yield unphysical predictions .  recently , a finite - volume discrete unified gas kinetic scheme ( dugks ) for molecule flows ranging from continuum to rarefied regimes has been developed , which has high accuracy and outstanding robustness . the nice asymptotic preserving ( ap ) properties also remove the restriction on the time step by the relaxation time that exists in other kinetic methods with direct discretization of the kinetic equation .furthermore , the finite - volume formulation enables the dugks to handle problems with complex geometries . some comparative studies suggest that the dugks has better performances over the lbm even for continuum flows . in this work , we will extend the dugks to phonon transport to construct an efficient method for the whole multiscale heat transport process ranging from diffusive to ballistic regimes .the remainder of the paper is organized as follows .section ii gives a brief introduction of the phonon bte , and the dugks for the bte is described in sec .some numerical simulations are carried out in sec .iv to test the scheme , and finally a brief summary is given in sec .in a rigid crystalline solids , the atomic vibrations from equilibrium positions can be quantized as quasi - particles known as phonons , and the system can be considered as a domain filled with a phonon gas .the angular frequency of a phonon is related to the wave number through certain dispersion relations , with different polarizations or modes of the phonon .the phonon transport can be described by the boltzmann transport equation in the regime as the wave effect of phonon is negligible , where ( or ) is the distribution function dependent on wave number ( or frequency ) , polarization , direction , and position as well as time ; is the group velocity with which the phonon of polarization travels .the term on the right hand side , , represents the rate of change of due to scattering interactions .usually the scattering is very complicated and a tractable model widely used is the relaxation time approximation , ,\ ] ] where is the relaxation time , is the equilibrium distribution of phonons and follows the bose - einstein distribution , with being the planck s constant divided by and the boltzmann constant , respectively , and is the temperature defined later .the effective relaxation time usually depends on temperature and frequency , and can be estimated using the matthiessen s rule if the individual scattering processes are independent of each other ,  where the relaxation times appearing on the right hand side are those due to the umklapp ( u ) and normal ( n ) phonon - phonon scatterings , boundary scattering , impurity scattering , etc . with the effective relaxation time, one can define the knudsen number of the system , , where is the characterize length of the system , and is the phonon mean free path with being a typical value of the phonon group velocity and a typical value of the relaxation time .the total energy and the heat flux can be defined from the phonon distribution function ,  where is the density of state , and is the solid angle .the temperature of the system can be obtained from , with being the volume specific heat capacity .even with the relaxation time approximation , the bte is still very difficult to be solved due to the high dimensionality .a number of tractable models have emerged to reduce the complex , such as gray model , semi - gray model , non - gray model , and phonon radiative transfer model. to illustrate the essence of our numerical method clearly without loss of generality , we will consider the gray model in the present work .this simplified model assumes phonons of all polarizations and frequencies are same , and the bte is expressed in terms of the phonon energy density  , ,\ ] ] where is the group velocity with being the magnitude , is the singlet relaxation time , and is the reduced distribution function for energy density , obviously , the total phonon energy and heat flux can be determined from , the equilibrium distribution function is just the angular average of the total energy ,   the gray model employs a single phonon group speed in all directions and a single relaxation time independent of polarization and frequency . despite the simple formulation ,the gray model can provide some insightful predictions on the phonon transport behaviors with acceptable accuracy . in the diffusive limit ( ) , it can be shown that the solution of the kinetic equation is determined by its average that obeys the diffusion equation ( see appendix a for details ) , where the thermal conductivity is given by we present the construction of the discrete unified gas kinetic scheme ( dugks ) for phonon transport based on eq . .first , the continuous solid angle domain is discretized into discrete angles using the discrete - ordinates method ( dom ) based on certain spherical quadratures , and correspondingly we obtain discrete directions .the accuracy of the quadrature employed is required to ensure the exact evaluation of the angular moments of the distribution function up to certain orders , such as            where and are the weights and the discrete angles of the corresponding spherical quadrature , is the discrete group velocity , and is the second - order unit tensor .these requirements suggest that the weights and the discrete angles satisfies with the discrete directions , the bte can then be expressed as for , where and .the total energy density and heat flux are then evaluated from the discrete distribution function for the phonon energy ,     the dugks method developed here is a finite - volume scheme for solving the bte , in which the spatial domain is divided into a set of control volumes .a one - dimensional ( 1d ) and two - dimensional ( 2d ) diagrams are shown in fig . [fig : cell ] as an example . integrating eq .( [ eq : bgk - phi ] ) in the volume centered at from time to leads to the following balance equation , ,\ ] ] where we have used the trapezoidal quadrature for the time integration of the collision term , and the mid - point rule for the flux term . is the volume of cell . here, is the flux across the interfaces of cell , where is the set consisting of the centers of cell interfaces , is the outward unit normal vector at of an interface , and is the corresponding interface area . in eq . , and are the cell - averaged values of the distribution function and collision term , respectively ,   the scheme ( [ eq : update_phi ] ) is implicit due to the inclusion of and the flux at half - time step . like in the original dugks for gas flows , the implicitness of the collision termcan be removed by introducing a new distribution function defined as or then eq .( [ eq : update_phi ] ) can be rewritten in terms of as where   note that the discrete scattering operator conserves energy , i.e , , therefore , we can track the evolution of instead of , and from the definition of , the energy and heat flux can be computed as where we have made use the fact that .now we discuss how to evaluate the cell interface flux at the half time - step , . to this end, we integrate eq . from to (here is the half time step ) along the characteristic line with the end point ( ) locating at the center of the cell interface ( see fig .[ fig : cell ] ) , ,\ ] ] where the trapezoidal rule is again used to evaluate the scattering term .the implicitness in this equation can be removed by introducing another auxiliary distribution function , or then we can obtain from eq .( [ eq : phi - face ] ) that with in order to determine , we assume that the distribution function is a piecewise linear function in the cell at which locates , say the cell centered at ( refer to fig . [ fig : cell ] ). then we can obtain that where is the slope of the distribution function in the cell centered at , which can be constructed smoothly or using certain numerical limiters .for instance , in 1d case as sketched in fig .[ fig : cell ] , the slope in cell can be determined by the central - difference , for smooth problems , or by the van leer limiter , \dfrac{|s_1||s_2|}{|s_1|+|s_2|},\ ] ] for problems with discontinuities , where it can be seen that in smooth region where , the slopes determined by the two methods are similar . for multi - dimensional case , the slope in each direction can be determined as described above .based on eqs .( [ eq : phi - bar ] ) and ( [ eq : smooth_phi+ ] ) , we can obtain that from which we can obtain the energy at the cell interface , where we have again used the energy conservative property of the discrete scattering operator. then the equilibrium distribution function can be obtained , and the original distribution function can be extracted from according to eq . , with the known distribution function at cell interface at the half time step , the flux can be evaluated according to eq .( [ eq : flux ] ) , and the cell - averaged distribution functions at the new time can be obtained according to eq . .summarizing the updating rule for the cell - averaged distribution functions and the interface flux , the dugks consists of the following two equations , +\dfrac{\delta t/2}{2\tau+\delta t/2 } \phi_{\alpha}^{eq , n+1/2}(\x_b),\ ] ] where ,\ ] ] .\ ] ] note that we have made use of eqs . and in the derivation of eq . .in practical computations , can be calculated from ,   the calculation procedure of the dugks at time step can be listed as follows :  a.  flux evaluation : * compute the auxiliary distribution functions according to eq . and its slope in each cell ; * compute the original cell interface distribution function according to eq . , where is evaluated based on given by eq . .b.  update of cell - averaged distribution functions : * compute the auxiliary distribution functions computed according to eq . ;* update the distribution functions in all cells via eq . .artificial diffusion can significantly deteriorate the simulation accuracy of a numerical scheme for the bte .we now analyze the numerical diffusion of the proposed dugks by analyzing the accuracy of the reconstructed cell - interface distribution functions .first it is noted that the exact solution of the bte at cell interface center can be written as the first and second terms on the right hand side represents the free transport and scattering processes , respectively . in the dugks, the trapezoidal rule is used to approximate the integral of the scattering term , and the approximation error in this term is ; for the first term on the right hand side , it is approximated by assuming the distribution function is a linear function in the cell , and the error is of order .therefore , the overall accuracy of the reconstructed distribution function at a cell interface in dugks is , and the numerical diffusion will also be of this order since the heat flux is the first - order angular moment of the reconstruction distribution function .note that the scattering term in eq .itself is of order , so if we neglect the scattering term totally ( i.e. only the free flight process is considered ) , like the classical first - order upwind ( step ) scheme , second - order upwind or central interpolation ( diamond ) scheme , the overall accuracy of the reconstructed distribution function will be , where the number depends on the employed interpolation rule .therefore , although the use of high - order interpolations can reduce numerical diffusion from spatial discretization , it is no help to reduce numerical diffusion from the scattering integration , which is of order . on the other hand ,it is known from eq . that the physical diffusion coefficient is proportional to the relaxation time .this suggests that in order to control the numerical diffusion to avoid false diffusion , it is required that . for problems in near ballistic regime ,this is not a problem since is relatively large .however , for diffusive and near diffusive problems , this becomes a rather severe limitation .therefore , for those explicit bte solvers that use direct interpolations , the small time step is required not only by the numerical stability condition , but also by the accuracy requirement .this also explains why some implicit bte solvers could produce large false diffusions even the computation is stable . on the other hand , the numerical diffusion from the discretization of the scattering term in present dugks is of order , which can release greatly the restriction on time step by the accuracy requirement .the asymptotic preserving ( ap ) property is important for a kinetic scheme in modeling multiscale transport .a kinetic scheme is ap given that ( i ) the time step is not limited by the relaxation time for any knudsen number , and ( ii ) the scheme is consistent with the free transport kinetic equation as , and consistent with the continuum equation as .  regarding point ( i ) , as discussed in the above subsection ,the restriction on the time step by accuracy requirement can be much released due to the coupling in the treatment of the scattering and transport processes .furthermore , the implicitness in the treatment of the collision terms with the trapezoidal rule in both the evolution of the cell - center distribution function [ eq . ] and the reconstruction of the cell - interface distribution [ eq . ] suggests that the restriction on the time step by numerical stability due to the relaxation time can be removed , too .therefore , the constraint on the time step of dugks due to the free flight process can be ensured by the courant - friedrichs - lewy ( cfl ) condition , where is the minimum cell size and is the cfl number .  to demonstrate point ( ii ) , we first rewrite eq . in terms of the original distribution function as \nonumber \\ & & + b(\tau , \delta t)\left[{\phi}_{\alpha}^{eq , n}(\x_c)+(\x'_{\alpha}-\x_c ) \cdot\bm{\delta}_c{\phi}_{\alpha}^{eq , n}\right ] + b(\tau , \delta t)\phi_{\alpha}^{eq , n+1/2}(\x_b).\end{aligned}\ ] ] where the two parameters and are note that in the dugks the discrete distribution functions are assumed to be piecewise linear in each cell , and therefore eq . can also be expressed as \nonumber \\ & & + b(\tau , \delta t)\left[{\phi}_{\alpha}^{eq , n}(\x_b)-\tfrac{\delta t}{2}\v_{\alpha } \cdot\bm{\delta}_c{\phi}_{\alpha}^{eq , n}\right ] + b(\tau , \delta t)\phi_{\alpha}^{eq , n+1/2}(\x_b),\end{aligned}\ ] ] where is the value of the distribution function at cell interface determined from cell .the ap property of the dugks depends on the parameters and .if we define the macroscopic diffusive length and time scales , and , respectively , such that and , the parameters and can the be expressed as where and are the dimensionless relaxation time and time step , respectively . in the ballistic limit ( ) , we can obtain that and , and thus which is just a solution of the free transport kinetic equation . specifically , in the 1d case as sketched in fig .[ fig : cell ] , the reconstructed interface distribution function is (v_\alpha ) + \left[{\phi}_{\alpha , j+1/2}^{n , r}-\tfrac{1}{2}v_{\alpha}\delta t \delta_{j+1}{\phi}_{\alpha}^{n}\right]\bar{h}(v_\alpha)\end{aligned}\ ] ] where and are the left and right values of the distribution functions at cell interface , respectively , is the heaviside function , if , and 0 otherwise , while .therefore , the dugks eq . can be expressed explicitly as ( note that in this limit ) +\bar{h}(v_{\alpha})\left[\phi_{\alpha , j+1/2}^{n , r}-\phi_{\alpha , j+1/2}^{n , r}\right ] \right\}\nonumber \\ & -&\dfrac{v_{\alpha}^2\delta t}{2\delta x_j } \left\ { h(v_{\alpha})\left[\delta_j\phi_{\alpha}^{n}-\delta_{j-1}\phi_{\alpha}^{n}\right ] + \bar{h}(v_{\alpha})\left[\delta_{j+1}\phi_{\alpha}^{n}-\delta_{j}\phi_{\alpha}^{n}\right ] \right\ } = 0,\end{aligned}\ ] ] which is a consistent finite - volume scheme of lax - wendroff type for the kinetic equation without scattering .this result suggests that the dugks is ap in the ballistic limit .now we discuss the ap property of the dugks in the diffusive limit ( ) . under this limit, the distribution function can be approximated as ( see appendix a ) , substituting this approximation into eq . we can obtain that where we have assumed that in diffusive limit the distribution function is smooth across cell interfaces , i.e. , and . on the other hand , based on the properties of the angular quadrature given by eq ., we have therefore , the macroscopic flux across the cell interface can be obtained , then by taking the zeroth angular moment of eq .we can obtain that specifically , for the 1d case , the above equation becomes = 0,\ ] ] which is just an explicit solver for the diffusion equation , suggesting that the dugks is also ap in the diffusive limit .generally three types of boundary conditions are used to describe the interactions between phonons and material boundaries , namely , specular reflection , thermalization , and diffusive reflection .the specular reflection is similar to that in classical gas kinetic theory .it assumes that a phonon is just reflected back to the domain with a reflection angle equal and opposite to the incident one after it hits the surface , such that the phonon energy density for directions entering the domain is given by where , with the outward unit normal vector to the wall pointing into the domain .therefore , for the specular reflection , the phonon energy is conserved and there is no energy transfer across the boundary .  in thermalization boundary condition ,a phonon is absorbed as it strikes the boundary , and a new phonon in thermal equilibrium with boundary temperature is emitted into the domain .therefore , the reflected phonon can be expressed as where is the temperature at the surface .since and are linear combinations of and , the thermalization boundary condition in the dugks can also be expressed as it can be seen that the thermalization boundary condition for phonon transport is very similar to the maxwell diffuse scattering in the classical gas kinetic theory , which assumes the distribution function of reflected particles follows a maxwellian one with the wall temperature and velocity .it is obvious that thermalization boundary condition allows for energy transfer across the surface .  in the diffusive reflection boundary condition , which should not be confused with the maxwell diffuse scattering in gas kinetic theory ,the phonons hitting the surface are reflected with equal probability along all possible angles , namely , ^{-1 } \int_{\bm{s}'\cdot\bm{n}<0 } { ( \bm{s}'\cdot\bm{n } ) f(\s')\,d\omega } , \ ] ] or in terms of the discrete phonon energy distribution function , ^{-1}\sum_{\s_{\beta}\cdot\bm{n } < 0}{w_{\beta}(\s_{\beta}\cdot\bm{n})\phi_{\beta}}.\ ] ] since the phonons hitting the surface are all reflected back to the domain , the total energy in the diffusive reflection boundary condition is conserved and no heat transfer occurs across the boundary .in this section we will apply the dugks to several heat transfer problems with different knudsen numbers to test its performance . in the simulations ,the cfl number is fixed at 0.9 unless stated otherwise .the van leer limiter is employed to determine the slope in each cell for problems as described in subsec .[ subsec : flux ] . the local coordinate system used in the simulationsis shown in fig .[ fig : coordinate ] , where  ] are the polar and azimuthal angles .     first we consider a dielectric film of thickness , the temperatures of the two boundaries located at and maintain at and , respectively .the problem is 1d and the energy distribution function depends on spacial variable and angular variable only , such that we can define a reduced distribution function by integrating with respect to , and the bte for the phonon energy density at steady state becomes ,\ ] ] where is the reduced equilibrium distribution for energy density . with the thermalizing boundary conditions at the two surfaces , the solution of this problem can be expressed as , ,\ ] ] where is the nondimensional position , is the nondimensional energy with and , is the acoustic thickness , and is the exponential integral function . the dimensionless heat flux can be expressed as which is a constant across the domain .the two integration equations and can be solved numerically to give an numerical exact " solutions . here we use 4000 points such that the solutions are mesh independent .the dugks is applied to this 1d problem at various knudsen numbers , and in the simulations the gauss - legendre quadrature with points is employed to evaluate the angular moments with respect to the directional cosine , i.e , it should be noted that here we choose such that the moments can be evaluated accurately for all of the knudsen numbers considered .actually , a quadrature with much less angular points can be employed as , say .the dimensionless energy heat flux with mesh resolutions of and 200 are shown in fig .[ fig : film ] .it can be observed that the dugks results agree well with the analytical solutions even with the coarse mesh , suggesting that the present scheme exhibits low numerical diffusion and is insensitive to mesh resolutions .the results also show the uniform stable property of the present dugks in the sense that the time step is solely determined by the cfl condition and is independent of the relaxation time .in other words , the time step ( or cell size ) is not required to be smaller than the relaxation time ( or mean free path ) .actually , with the coarse mesh ( ) , the value of ranges from 0.001 to 100 as changes from to , and ranges from to 90 .  for comparison, we also applied the implicit finite - difference method with step ( upwind ) discretization of the convection term of eq ., which is widely used for solving the phonon bte , to the present problem .the profiles of the dimensionless energy at various kn are shown in fig .[ fig : step ] .it can be observed with the fine mesh of , the results agree well with the analytical solutions as changes from 0.01 to 10 .this is reasonable since in all of the considered cases , and the numerical diffusion , which is proportional to , is less than the physical one with this mesh resolution . on the other hand , with the coarse mesh , clear deviations from the analytical solutions can be observed as and 0.2 .     against dimensionless time at different knudsen numbers .( a ) finite values of ; ( b ) diffusive limit ; ( c ) ballistic limit . ,title="fig:",scaledwidth=60.0% ] against dimensionless time at different knudsen numbers .( a ) finite values of ; ( b ) diffusive limit ; ( c ) ballistic limit . ,title="fig:",scaledwidth=45.0% ] against dimensionless time at different knudsen numbers . ( a ) finite values of ; ( b )diffusive limit ; ( c ) ballistic limit . , title="fig:",scaledwidth=45.0% ]   the transient thermal grating ( ttg ) is a technique for measuring thermal conductivity and phonon mean free path of a material . here we considered the thermal relaxation process of a 1d ttg , where initially two crossed laser pulses are imposed to produce a spatially sinusoidal temperature variation in the medium , where is the background temperature , is the amplitude of the temperature variation , and is the wave number with being the grating period . as the strength of the pulse is weak , the bte can be linearized , and the temperature deviation from the background temperature , , can be approximated as , where the amplitude can be obtained analytically , ^{(t'-t^*)}\ , dt'},\ ] ] with and , and is the rarefaction parameter with the knudsen number defined as .equation is a second kind volterra integral equation and can be solved using standard numerical techniques . in the diffusive limit ( ) , it can be shown that where ; while in the ballistic limit ( ) , we have , which shows strong oscillations .we simulate the thermal relaxation process of the 1d ttg at different knudsen numbers .the length of the computation domain is taken to be , and a uniform grid of 100 points ( i.e. , 50 points in one grating period ) is used for all cases .the directional cosine space is discretized using the gauss - legendre quadrature with points .periodic boundary conditions are applied to the left ( ) and right ( ) boundaries . in fig .[ fig : ttg - t ] the time histories of the amplitude of the temperature variation are shown for different values of .it can be seen that the numerical results of the dugks are in excellent agreement with the analytical solutions for different knudsen numbers , ranging from diffusive limit to ballistic limit .the results again confirm the asymptotic preserving properties of the present dugks method for modeling multiscale heat transfer .   in the above subsectionsthe dugks was tested by 1d steady and 1d unsteady heat transfer problems .we now consider a two - dimensional problem in a square medium of length .initially the temperature of the medium is set to be a uniform , and then the temperature at the bottom side ( ) is raised to and maintained at .thermalization boundary conditions are assumed on the four sides , and the knudsen number of the system is defined as .this problem was studied recently by solving the bte with a dom coupled with finite - element scheme , and a similar radiative heat transfer was studied early , which is actually identical to the phonon transfer as the medium is exposed to isotropically scattering .simulations at different knudsen numbers are conducted based on a uniform mesh in physical space , and the directional cosine  ] ( not $ ] due to symmetry ) is discretized with the 16-point gauss - legendre quadrature .our simulations show that the meshes are sufficient to obtain convergent results .  in fig .[ fig:2d ] the normalized temperatures , , are shown along the vertical centerline at as the system reaches the steady state .for comparison , we also present the solutions of dom for phonon bte and the numerical analytical solutions of the integral equation describing radiative transfer in a 2d isotropically scattering medium . it can be seen that the results of the dugks agree quite well with the reference data for all knudsen numbers considered here .it is noted that some slight oscillations appear in the temperature profile predicted by the dom at . this is caused by the ray effect due to the insufficient discretization in angular space , which was based on a gaussian quadrature . on the other hand , the temperature profile from the present dugks does not show this behavior with the angular discretization .   and[ ] , scaledwidth=60.0% ]  the transient behavior of this 2d problem is also compared with the results of the dom . the temperature profiles along the vertical centerline of the medium are shown in fig .[ fig : time2d - t ] at different times of for different knudsen numbers ranging from diffusive to ballistic regimes .it can be observed that the dugks results agree qualitatively well with the dom predictions from initial to later times in all cases considered .specifically , as , the heat transfer is diffusive , and no obvious temperature jump occurs at the top and bottom boundaries at different times .as kn increases to 0.1 , i.e. , the problem falls in the near diffusive regime , and temperature jump appears on both boundaries , particularly on the bottom hot surface , where the jump decreases with time . at ,temperature jump is more significant , but transient jump becomes smaller than that for the case of . as , the heat transfer is dominated by ballistic effect , and the temperature jump maintains nearly constant on the hot surface , although transient changes can still be observed on the bottom cold surface .it is also noted that some differences between the present results and the dom data for the case of at , which can again be attributed to the insufficient angular discretization of the dom .overall , the results of this 2d heat transfer problem confirm the ap and uniform stable properties of the proposed dugks for simulating heat transport process from diffusive to ballistic regimes .  .,title="fig:",scaledwidth=45.0%].,title="fig:",scaledwidth=45.0%].,title="fig:",scaledwidth=45.0%].,title="fig:",scaledwidth=45.0% ]in this work we develop a discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon boltzmann transport equation . with the coupled treatment of the scattering and transport of phonons ,the scheme is asymptotic preserving , and the limitation on the time step being less than the relaxation time in diffusive regime is removed .the method has been validated in two 1d and a 2d problems at different knudsen numbers , and the results are in excellent agreement with analytical or benchmark data .  in the present work ,the proposed dugks is designed based on the gray bte model , where the frequency dependence of the distribution function , relaxation time , and group velocity , are ignored .however , the current methodology can be extended to non - gray bte models , which will be studied in our subsequent works .in order to analyze the limit behavior of the bte in the diffusive regime , we here make use of the diffusive scaling analysis introduced by sone in gas kinetic theory , which was also used in the analysis of the unified gask kinetic scheme for radiative transper problems . with the diffusive scaling , i.e. , and with being a small parameter , we have then the bte can be rewritten as .\ ] ] we further expand in a power series of , substituting this expansion into eq . , we can obtain the equations in the consecutive orders of ,   from eqs . and , and recalling the energy conservative property of the scattering operator, we have that then taking moment of eq . we can obtain the following macroscopic equation , where , which can be obtained from eq . as which is exactly the fourier law .therefore , the heat transfer equation at the ( macroscopic ) diffusive length and time scales are or with being the heat conductivity .  with the equations at different orders of given by eq ., we can obtain an approximation of the energy distribution function at the first order of , which is used in the discussion on the ap property of the dugks in the diffusive limit , i.e. , eq . .70 j. y. murthy , s. v. j. narumanchi , j. a. pascual - 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Abstract: numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved . in this work a discrete unified gas kinetic scheme ( dugks ) is developed for heat transfer in materials with different acoustic thickness based on the phonon boltzmann equation . with discrete phonon direction , the boltzmann equation is discretized with a second - order finite - volume formulation , in which the time - step is fully determined by the courant - friedrichs - lewy ( cfl ) condition . the scheme has the asymptotic preserving ( ap ) properties for both diffusive and ballistic regimes , and can present accurate solutions in the whole transition regime as well . the dugks is a self - adaptive multiscale method for the capturing of local transport process . numerical tests for both heat transfers with different knudsen numbers are presented to validate the current method .


molecular dynamics simulations are a powerful tool for determining the behavior of multiparticle systems and are used in a wide range of applications .there are two basic approaches to these simulations :  \(i ) time driven simulations in which equations of motion of all particles are solved for a series of small time slices .the positions and velocities of the particles are determined at the end of each time slice and used as input to the calculation for the next time slice .\(ii ) event driven simulations which are applicable to systems of hard spheres or more generally to systems with interparticle potentials which are piecewise constant .the approach with event driven simulations is to determine when the next collision between two particles occurs , determine the positions and velocities of these particles after the collision and then repeat this process .a collision is defined as the event in which the hard spheres collide or more generally when two particles reach a discontinuity in their interparticle potential .we focus here on event driven simulations which , where applicable , provide exact results and typically run faster than time driven simulations .determination of the next event is usually composed of two steps :  \(i ) determination of the collision event with the shortest time for each particle . by dividing the system into cells and/or maintaining lists of particles within a certain distance of a given particle ( neighbor lists ) , the time taken for calculation of the first collision event for a given particle can be made independent of n , the total number of particles in the system .\(ii ) determination of the collision event with the shortest time among all the particles , given the events with the shortest time for each particle obtained in ( i ) .approaches have been proposed and implemented which allow this determination in o( ) time .the subject of this paper is an approach to determining the next collision event among all particles .this has been a heavily researched subject .the requirements for a queue to allow this determination is as follows .the queue must support :  \(i ) addition of an event to the queue ;  \(ii ) identification and deletion from the queue of the event with the shortest collision time ;  \(iii ) deletion of a given event from the queue ( e.g. when a collision ( p , q ) occurs we may want to remove the event ( q , p ) from the queue . )these requirements define abstractly the concept of a _ priority queue_.  implementations of priority queues for molecular dynamics simulations have for the most part been based on various types of binary trees .they all share the property that determining the event in the queue with the smallest value requires o( ) time .the early work on priority queues is reviewed in ref .the earliest implementations of priority queues used linked lists which results in performance .implementations with performance were introduced and analyzed in refs .the oldest priority queue implementations with performance used _ implicit heaps _ binary trees in which each item always has a priority higher than its children and the tree is embedded in an array .other implementations include _ leftist trees _ , _ binomial queues _ , _ pagodas _ , _ skew heaps _ , _ splay trees _ and _ pairing heaps _ .marin et al . introduced a version of the _ complete binary tree _ which also has performance and compared it to earlier priority queue implementations explicitly in the context of molecular dynamics simulations . they find that over a wide range of densities their complete binary tree variant has the best performance in terms of the coefficient of the term and large behavior .  in this work, we propose a priority queue for use in event driven molecular dynamics simulations for which all operations require o(1 ) time .the approach is inspired by the concept of a _ bounded priority queue _ which is typically implemented as an array of linear lists and which is applicable to problems in which the values associated with queue items are integers and are bounded ( i.e. the values associated with events obey where and are constants ) . bounded priority queues are not directly applicable to the molecular dynamics queueing problem because neither of these requirements are met .we show , however , that with a hybrid approach that employs both a normal priority queue and a bounded priority queue we can ensure all operations on the queue take o(1 ) time .we make use of the facts that for molecular dynamics simulations :  \(i ) the time associated with an event to be added to the queue is always later than the time associated with the last event removed from the top of the queue .that is , where is the time associated with the event to be added to the queue and is the last event removed from the top of the queue .\(ii ) there exists a constant , such that we call a priority queue which supports such events a bipq ( bounded increasing priority queue ) .the basic idea is to :  \(i ) perform a gross sort of the events using an array of linear lists and  \(ii ) to use a binary tree to perform a fine sort of only those events which are currently candidates for the event with the shortest time .more specifically , our priority queue is composed of the following components :  \1 .an array , , of linear lists , .( section [ choose ] below discusses how to determine the the size of the array . )the array is treated in a circular manner .that is , the last linear list in the array is followed logically by the first linear list .we implement each linear list as a doubly linked list .\2 . a binary tree which is used to implement a conventional priority queue .we also maintain two additional quantities : the _ current index _, , and a _ base index _ associated with the queue .initially , all linear lists and the binary tree are empty and and are .here we describe how operations on the queue are implemented using the data structures described above .\(i ) addition .events are added to either one of the linear lists or to the binary tree as follows : an index for the event to be added is determined by where : is the time associated with the event ; is the base index ; and is a _ scale factor _ the value of which is such the binary tree never contains more than a relatively small number of events ( ) .if is equal to the _ current index _ , , the event is added to the binary tree , otherwise it is added to linear list .\(ii ) identification of the event with shortest time .the event with the shortest time is simply the root of the binary tree , as is the case with a normal priority queue implemented using a binary tree .if a request is made for the event with the smallest time and the binary tree is empty , the current index is incremented by one ( wrapping around to if we reach the end of the array ) and all events in the linear list inserted in the binary tree .if there are none , we continue to increment until a non - empty linear list is found .if we wrap around to the beginning of the list , is incremented by .we find that in practice , when the binary tree becomes empty the next linear list is always non - empty ( see section [ choose ] in which we show the distribution of event times ) .\(iii ) deletion of an event .we simply delete the event from the array of linear lists or from the binary tree depending on the structure in which it is located .the fact that the time associated with an event to be added to the queue is always greater than or equal to the time associated with the last event removed allows us to use the array of linear lists in a circular fashion .the requirement that there exists a constant , such that allows us to use a finite number of linear lists .the number of linear lists required is proportional to . in practicewe find that we can always find a reasonable value of such that eq .( [ tmax ] )  holds . if a rare event occurs which violates this constraint or we want to use less memory for linear lists causing the constraint to be violated , the event is handled on an exception basis as implemented in the _ processoverflowlist _ function in code contained in the appendixalternatively , the application which calls the priority queue code can guarantee that such an event never occurs by creating an earlier fictitious collision with a time which does not violate the constraint .  thus all of the events , except for those deleted before they are placed in the binary tree , will eventually be added to the binary tree , but at any given time the tree , instead of containing o( ) entries , will contain only a relatively small number of entries .the number of events maintained in the binary tree is only a fraction of the total number of particles in the system and can be made independent of .our priority queue is similar to a _calendar queue _ ; however , the calender queue does not employ a binary tree events are sorted in each of the linear lists .two parameters , the number of linear lists and the scale factor , must be chosen to specify the implementation of the queue .operationally , they can be chosen as follows :  \(i ) first , by instrumenting the queue to count the number of events in the binary tree , determine a value of such that the number of events in the binary tree is relatively small ( ) .table i. and fig .[ mem](a ) summarize the values of we have used for our simulations .the figure is consistent with a scale factor linear in with a different coefficient of linearity dependent on density . because we use a binary tree to store events with the soonest times , the performance of the algorithm is somewhat insensitive to the choice of .for example , a choice of which results in a doubling of the number of events in the binary tree results in only one additional level in the tree .\(ii ) instrument the queue to find , the maximum difference between the time associated with an event to be added and the time associated with the last event removed and set to ensure that ( eq . [ tmax ] ) is met .table i. and fig .[ mem](b ) summarize the number of linear lists we have used for our simulations . as with , is linear in with a different coefficient of linearity dependent on density .we note that while memory requirements are as in the conventional implementation of priority queues , the hybrid implementation does require significantly more memory than the conventional implementation due to the memory required for the linear lists .tradefoffs can be made of cpu time for memory by increasing the scale factor and/or reducing the number of linear lists ( resulting in more exception conditions ) .figure  [ pdist](a ) plots the average number of events with index versus for various . here that is , is the distance of from the current index taking into account the circular nature of the array of linear lists .the data was obtained by sampling the queue many times at regular intervals . with the choice of scale factors shown in table iwe achieve our goal of having events with index and thus in the binary tree .note that to achieve this , the scale factor increases with increasing resulting in the cutoff of the distributions also increasing with increasing .( in fact , if the -axis is transformed by , the plots collapse as shown in fig .[ pdist](b ) reflecting the fact that the probability distribution of collision times is independent of . )thus , the number of linear lists required to ensure that eq .( [ tmax ] ) holds also increases with . in fig .[ pbucket0 ] , we plot the distribution the probability that the number of events with the current index , is versus .the distributions are strongly peaked indicating that the number of events in the binary tree do not vary much from the average .the basic operations involved in the queue are :  \(i ) insertion into and deletion from the linear lists .use of doubly linked lists allows these operations to be implemented to take o(1 ) time .\(ii ) binary tree operations .we use the code of ref . to implement the binary tree operations .when a leaf representing an item in the priority queue is added to or deleted from the tree , the tree must be traversed from the affected leaf possibly all the way to the root node and adjustments made to reflect the presence or absence of the affected leaf .thus a bound on the number of levels which must be traversed is where is the number of items in the priority queue . in sec .[ choose ] we show that by choosing the scale factor appropriately , can be made to be independent of ( and have a relatively small value , ) .thus binary tree operations will be o(1 ) .\(iii ) identification of the next non - empty linear list , after the current linear list is exhausted . as explained in item ( ii ) of sec .[ queueoperation ] , when the binary tree is empty , we search forward through the array of linear lists until a non - empty list is found . if the number of lists we must search through increases with , this process will not be o(1 ) .we show below that with the proper choice of , the number of lists we must search does not grow with and in fact show that the next linear list after the current one almost always is non - empty .thus the complexity of identification of the next non - empty list will be o(1 ) .thus the overall time taken by queue operations per collision is o(1 ) .we run simulations using both a conventional priority queue and our new hybrid approach . for simplicitythe simulation was of identical size hard spheres of radius one and unit mass .the sizes of the cubic systems are set to maintain equal densities .the parameters of the simulation are as shown in table i.  to demonstrate the performance of our approach , we run simulations for cubic systems at four volume densities and . the first density represents a rarefied gas and the last density represents a jammed system .the jamming density for hard sphere systems is  .for both the conventional priority queue and the hybrid queue we used the binary tree code from ref . .figure [ pcbt ] shows the time taken for collisions for queue operations with both a conventional priority queue and the hybrid queue .as expected , the time for the conventional priority queue increases as while the time for the hybrid queue is essentially constant.  there is , however , a slight upward trend in the hybrid queue results . to determine if this trend is a feature of the algorithm or of the the benchmark environment , we proceed as follows .we first study the only two places in the hybrid code where looping is involved :  \(i ) in the _ updatecbt _ function of ref . we loop as we traverse the binary tree .if we traverse more levels as the n grows , the algorithm will not be o(1 ) . to explore this possibility , we instrument the function to count the number of number of levels we traverse in the tree . the results are shown in fig .the number of loop iterations is essentially constant , independent of .\(ii ) in the _ deletefirstfromeventq _ function , after the priority queue for the current linear list is exhausted , we loop until we find a non - empty list .if the number of lists we must examine before we find the first non - empty list grows with the system size , the algorithm will not be o(1 ) .we examine this possibility by counting the number of times we encounter an empty list and find that on average the probability of encountering an empty list does not grow with and that the probability of encountering an empty list is very small : we encounter an empty list only of the times after exhausting the priority queue .having ruled out dependence of the number of loop iterations on as the source of the upward trend in the execution times , we now consider whether the larger memory needed as increases is the cause of the trend .all modern computer processors employ high speed cpu cache memory to reduce the average time to access memory .in fact , the processor we use in our simulations , the amd opteron , employs a two - level memory cache ( 64 kb level 1 cache , 1 mb level 2 cache ) .a similar cache structure is used in the intel xeon processor . because memory caches are finite size , if the memory access is random the larger the memory used by a program , the lower the probability that data will be found in the cache resulting in slower instruction execution .the effect of cache in benchmark runtimes has been studied in ref .we study the effect of the finite size of the cache in our system as follows : instead of running the molecular dynamics simulations , we run a small test program which randomly accesses the data structures used by the molecular dynamics simulations . for each value of , the test program executes exactly the same number of instructions but uses data structures of the size used by the molecular dynamics simulations for that value of . the results are shown in fig .[ pcbt ] and show an upward trend similar to that of the simulation results for all of the densities studied .the above results thus suggest that the complexity of the hybrid algorithm is , in fact , o(1 ) and that the upward trend in the results is due to the finite size of the high speed memory cache .we have defined a new abstract data type , the bounded increasing priority queue ( bipq ) having the same operations as a conventional priority queue but which takes advantage of the fact that the value associated with an item to be added to the queue has the properties that : ( i ) the value is greater than or equal to the value associated with the last item removed from the top of the queue and ( ii ) the value minus the value of the last item removed from the top of the queue is bounded .these properties are obeyed for events in event driven molecular dynamic simulations .we implement a bipq using a hybrid approach incorporating a conventional priority queue ( implemented with a binary tree ) and a bounded priority queue .all operations on the bipq take an average o(1 ) time per collision .this type of queue should provide performance speedups for molecular dynamics simulations in which the event queue is the bottleneck .we thank sergey buldyrev , pradeep kumar , sameet sreenivasan , and brigita urbanc for helpful discussions .we onr , nsf and nih for support .the following code implements the hybrid queue proposed here .the calls to insert and delete are to the functions contained in ref . , which update np and the complete binary tree , cbt .any code providing the same functions could be substituted for insert and delete .i=(int)(scale*pt->t - baseindex ) ;  if(i>(nlists-1 ) )   / * account for wrap * /        {   i-=nlists ;   if(i>=currentindex-1 )   {       i = nlists ; / * store in overflow list * /   }  }  pt->qindex = i ;      processoverflowlist ( ) {  int i , e , enext ;  i = nlists ;  / * overflow list * /  e = linearlists[i ] ;  linearlists[i]=-1 ;  / * mark empty ; we will    treat all entries and may re - add some * /  while(e!=-1 )  {   enext = eventqentries[e].next ; / * save next* /   insertineventq(e ) ;  / * try add to regular list now * /   e = enext ;  } }                           j. j. erpenbeck and w. w. wood , in statistical mechanics b : modern theoretical chemistry , b.j.berne ( ed . ) , molecular dynamics techniques for hard core systems , vol.6 , institute of physics publishing , london , ( 1977 ) 1 - 40 .j. francon , g. viennot , and j. vuillemin , description and analysis of an efficient priority queue representation . in proceeding of the 19th annual symposium on foundations of computer science .ieee ( 1978 ) 1 - 7 .

Abstract: we propose and implement a priority queue suitable for use in event driven molecular dynamics simulations . all operations on the queue take on average o(1 ) time per collision . in comparison , previously studied queues for event driven molecular dynamics simulations require o(log ) time per collision for systems of particles .