Simulating radiative transfer with filtered spherical harmonics

This Letter presents a novel application of filters to the spherical harmonics (PN) expansion for radiative transfer problems in the high-energy-density regime. The filter, which is based on non-oscillatory spherical splines, preserves both the equilibrium diffusion limit and formal convergence properties of the unfiltered expansion. While the method requires further mathematical justification and computational studies, preliminary results demonstrate that solutions to the filtered PN equations are (1) more robust and less oscillatory than standard PN solutions and (2) more accurate than discrete ordinates solutions of comparable order. The filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem. Given the benefits of this method we believe it will enable more routine use of high-fidelity radiation-hydrodynamics calculations in the simulation of physical systems.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

We propose the P_N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P_N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.     

Second-order time evolution of PN equations for radiation transport

Using polynomials to represent the angular variation of the radiation intensity is usually referred to as the P_N or spherical harmonics method. For infinite order, the representation is an exact solution of the radiation transport solution. For finite N, in some physical situations there are oscillations in the solution that can make the radiation energy density be negative. For small N, the oscillations may be large enough to force the material temperature to numerically have non-physical negative values. The second-order time evolution algorithm presented here allows for more accurate solutions with larger time steps; however, it also can resolve the negativities that first-order time solutions smear out. Therefore, artificial scattering is studied to see how it can be used to decrease the oscillations in low-order solutions and prevent negativities. Small amounts of arbitrary, non-physical scattering can significantly improve the accuracy of the solution to test problems. Flux-limited diffusion solutions can also be improved by including artificial scattering. One- and two-dimensional test results are presented.     

The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictable zone

We have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rate ? is smaller than 0:11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate “2C _P ”[7], close to the second laser threshold “2C _2th ”(threshold of instability). When the pumping parameter “2C” increases, the laser undergoes a period doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and fifth order harmonics for the laser-field expansion [7, 17]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

Solution of the three-dimensional problem of wave diffraction by an ensemble of objects

The pattern equations method is extended to solving three-dimensional problems of wave diffraction by an ensemble of bodies. The method is based on the reduction of the initial problem to a system of N (N is the number of scatterers in the ensemble) integro-operator equations of the second kind for the scattering patterns of scatterers. With the use of the series expansions of the scattering patterns in angular spherical harmonics, the problem is reduced to an algebraic system of equations in the expansion coefficients. An explicit (asymptotic) solution to the problems is obtained in the case when the scattering bodies are separated by sufficiently long distances. It is shown that the method can be used to model the characteristics of wave scattering by complex-shaped bodies.     

Magnetic black holes with string-loop corrections

String-loop corrections to magnetic black holes are studied. 4D effective action is obtained by compactification of the heterotic string theory on the manifold K3×T² or on a suitable orbifold yielding N=1 supersymmetry in 6D. In the resulting 4D theory with N=2 local supersymmetry, the prepotential receives only one-string-loop perturbative correction. The loop-corrected black hole is obtained in two approaches: (i) by solving the system of the Einstein-Maxwell equations of motion derived from the loop-corrected effective action and (ii) by solving the system of spinor Killing equations (conditions for the supersymmetry variations of the fermions to vanish) and Maxwell equations. We consider a particular tree-level solution with the magnetic charges adjusted so that the moduli connected with the metric of the internal two-torus are constant. In this case, the loop correction to the prepotential is independent of coordinates, and it is possible to solve the system of the Einstein-Maxwell and spinor Killing equations in the first order in string coupling analytically. The set of supersymmetric solutions of the loop-corrected spinor Killing equations is contained in a larger set of solutions of the equations of motion derived from the string-loop-corrected effective action. Loop corrections to the metric and dilaton are large at small distances from the center of the black hole.     

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

Some comments on the rational solutions of the Zakharov-Schabat equations

We describe a family of the rational solutions of the Zakharov—Schabat equations. This family is characterized by extremely simple superposition principle, following directly from the Darboux-invariance of the Zakharov-Schabat equations proved in the works [1, 4]. Particularly we present an infinite sequence of polynomials P _n (x, y, t, t ₄, ..., t _m), m≤n, so that the formula $$u = 2\partial _x^2 Log\left( {\sum\limits_{i = 1}^N {c_i P_i } } \right)$$ where c _i are the arbitrary constants, represents some class of solutions of the Kadomtcev—Petviashvily equation. The paramters t ₄, ..., t _K represent the explicit action of the commuting flows, related with the Zakharov—Schabat operators of the higher order, on the solutions of the K—P equation, and can be fixed independently in each P _i. The polynomials P _n are closely related with the second Waring formular well known in algebra. This relation imposes some specific constraints on the motion of the N particle Moser—Calogero system generated by P _n.     

Light scattering by multilayer axially symmetric particles

An exact solution to the problem of light scattering by multilayer axially symmetric particles is derived and some aspects of its computer-aided implementation are discussed. The main specific features of the solution are (i) separation of the incident, scattered, and internal fields into two parts and special selection of the scalar potentials for each of them; (ii) expansion of the potentials in terms of spherical wave functions; (iii) formulation of the problem in the form of surface integral equations; and (iv) solution of the reduced systems of the linear algebraic equations for the coefficients of the potential expansions. Mathematical justification of the solution is discussed, which is formulated in the recursive and nonrecursive form (for the T-matrix). The developed computer program has shown that the proposed approach makes it possible to consider axially symmetric particles with essentially different internal structures (i.e., with a spherical core, oblate spheroidal shell, or prolate spheroidal intermediate layer). The results of calculations of the optical properties of the multilayer nonspherical particles are presented and discussed.     

Comparison of radiative transfer approximations for a highly forward scattering planar medium

We examine critically the accuracy of the two-flux, spherical harmonics and discrete ordinates methods for predicting radiative transfer in a planar, highly-forward scattering and absorbing medium. Numerical results for the radiative fluxes show that the two-flux and P₃-approximations yield accurate results compared to solutions based on the F_N-method. Indeed, these approximate methods are relatively simple and have potential for generalization to predict radiative transfer in multidimensional systems, as long as an appropriate simplification of the phase function is utilized.     

Solid-liquid equilibria for quaternary solid solutions involving compound semiconductors in the regular solution approximation

Liquidus equations for solid-liquid equilibria in quaternary systems involving compound semiconductors are presented in both thermodynamic (model-independent) and regular solution forms. A unified treatment is offered for terminal (doubly-doped binary compound, singly-doped ternary solution) as well as continuous series of solid solutions (mixing on one or both sublattices) which is facilitated by the choice of binary compounds as components in the solid. With the exception of the quaternary solid solution with mixing on both sublattices (i.e. Al_uGa_1?uP_vAs_1?v), the liquidus equations are generalizations of previous results for ternary systems. In the case of mixing on both sublattices, only three of the four possible liquidus equations are independent on account of the equilibrium among the four compound components. The required binary compound activities in the quaternary solution are deduced from a statistical treatment of a regular mixture of four kinds of atoms subject to the site restrictions of the zinc-blende lattice. The calculation of quaternary phase diagrams from binary and ternary data is discussed. Finally, it is found for the case of mixing on both sublattices that previous results, based on the decomposition of the quaternary solid solution into a binary regular mixture of ternary solids, are consistent with the present analysis.     

Time-dependent diffusion and transport calculations using a finite-element-spherical harmonics method

In this paper, the finite-element-spherical harmonics (FE-P_N) method is applied to the solution of transient Boltzmann transport equation. Firstly, transport and diffusion calculations are obtained for homogeneous and inhomogeneous circular regions. Results are compared in order to show the effects of different absorption coefficient values on the propagation of photons. Significant differences between two theories are shown to occur especially in cases when the absorption is increased. Secondly, to validate the FE-P_N method, results from this method are compared with Monte Carlo calculations for different cases. Comparisons show good agreements between FE-transport and Monte Carlo solutions and demonstrate the correctness of the results obtained.     

Eigenvalue spectrum with chebyshev polynomial approximation of the transport equation in slab geometry

Using certain well-known properties of chebyshev polynomials, a simple and highly efficient approach to evaluate eigenvalue in radiation transport is presented. The spectrum of eigenvalues has been studied for slabs with isotropic scattering of different magnitudes of the cross section parameter c (i.e., the mean number of neutrons emitted per collision). It is shown that in the presence of the chebyshev polynomial approximation (T_N) there are both discrete and continuum of eigenvalues. It is found that the T_N method gives very good agreement with conventional spherical harmonics approximation (P_N).     

无限长黑圆柱情形下密恩(Milne)问题的近似解(球谐函数展开法)

本文探讨当一无限长黑圆柱放在一满足密恩问题中诸条件的无限介质中时,介质中的中子分布。计算采用球谐函数展开法,把中子分布函数对球谐函数展开,保留展开式的起首若干项,从而求得近似解。具体计算作到P₅近似为止。表2及附图示各次近似中对於圆柱半径α的不同值求出的外推长度λ之值。作为长度单位的是中子在介质中的平均自由路程l。为比较起见,我们在图中也画出了达维逊(Davison)给出的曲线(曲线D)。他的曲线是根据α《1及α》1二极限情形下派耳斯(Peierls)积分方程的近似解,中间参照P₃近似的结果画出的。由图可见,α大时P₅近似的结果已很接近於曲线D,而在α=1附近,则曲线D似乎远应该略低一些,才更符合曲线P₅的趋势(例如,像图中虚线所表示的那样)。     

QCD nuclear factor and moments of multiplicity distributions in higher orders of perturbation theory

Perturbative solutions to the gluodynamics equations withmodified integral kernel in the second and third orders of the perturbation theory (2NLO, i.e., next-to-next-to-leading order, and 3NLO, i.e., next-to-next-to-next-to-leading order approximations) were found. The effect of the nuclear factor N _S on the behavior of the ratio H _q of cumulant and factorial moments of multiplicity distributions was analyzed. Theoretical conclusions were compared to the results of simulation of p-Pb and Pb-Pb collisions of nuclei with energies of 200 and 546 GeV/nucleon.     

Analytic approach to the orientational ordering in the high pressure phase II of o-D2 and p-H2

Possible kinds of orientational order in the high pressure phase II of o-D₂ and p-H₂ are considered as bifurcations of solutions of some nonlinear equations. Before, we found this approach to be fruitful in the case of orientational ordering in o-H₂ and p-D₂. A solution with four sublattices in the hcp structure is obtained. The order parameter of this solution abruptly decreases with temperature and becomes zero in phase I. The curve of the phase boundary T_c(P) is calculated.     

A generalization of the Virasoro algebra to arbitrary dimensions

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.     

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

We study the semiclassical limit of the (generalized) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in H ^s to the solution of the Hopf equation, provided the initial data belongs to H ^s , ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.