error analysis and comparative study of numerical methods for the parabolic equation applied to tunnel propagation modeling parabolic equation pe methods have been widely applied to the modeling of wireless propagation in tunnel environments however the relevant literature does not include concrete guidelines for the choice of the parameters of these methods and the tradeoffs involved this paper provides a comprehensive analysis of the two sources of error that arise when pe methods are employed for the modeling of radio wave propagation scenarios the well known numerical dispersion error stemming from the finite difference solvers for pe and the approximation error stemming from the use of pe for the solution of wave propagation problems that are subject to maxwell s equations the analysis is performed for four methods three of which have been already used in pe based propagation studies namely the crank nicolson cn scheme the alternative direction implicit adi method and its locally one dimensional lod adi version the fourth method is the mitchell fairweather mf adi scheme that has been recently shown to be a promising alternative technique for tunnel propagation modeling the proposed method leads to robust criteria for the choice of spatial discretization in realistic propagation scenarios as shown via numerical examples

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

preconditioned krylov subspace methods for solving radiative transfer problems with scattering and reflection abstract two krylov subspace methods the gmres and the bicgstab are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation to increase their convergence rate and stability the jacobi and block jacobi methods are used as preconditioners for both krylov subspace methods numerical experiments designed to test the effectiveness of the preconditioned gmres and the bicgstab are performed on various radiative transfer problems i transparent ii absorption dominant iii scattering dominant and iv with specular reflection it is observed that the bicgstab is superior to the gmres with lower iteration counts solving times and memory consumption in particular the bicgstab preconditioned by the block jacobi method performed best amongst the set of other solvers to better understand the discrete systems for radiative problems i to iv an eigenvalue spectrum analysis has also been performed it revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems this conditioning further deteriorates when reflection is involved

a fast multilevel algorithm for the solution of nonlinear systems of conductive radiative heat transfer equations in this paper we describe and analyze a fast multilevel algorithm for the solution of a system of nonlinear integro differential equations that model steady state combined conductive radiative heat transfer this system of equations for radiative intensity and temperature can be formulated as a compact fixed point problem in temperature alone with a fixed point map that requires both a solution of the linear transport equation and the linear heat equation for its evaluation we obtain an efficient evaluation of the fixed point map by coupling a finite element diffusion solver with a fast transport solver developed by the second author as a solver we apply a modification of the atkinson brakhage method with newton gmres as the coarse mesh solver to the full nonlinear system we compare our discretization solver pair with newton gmres and the classical atkinson brakhage algorithm

jsj splittings for finitely presented groups over slender groups we generalize the jsj splitting of rips and sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups such classes include the class of all 2 ended groups and the class of all virtually z z groups the approach called track zipping is relatively elementary and differs from the rips sela approach in that it does not rely on the theory of r trees but rather on an understanding of certain embedded 1 complexes called patterns in a presentation 2 complex for the ambient group

theoretical analysis of an optimal control problem of conductive convective radiative heat transfer abstract the problem of optimal heat removal from a three dimensional domain is considered the specific of the study consist in accounting for the radiative heat transfer the so called p1 approximation of the radiative heat transfer equation is used which reduces the model to a nonlinear elliptic system a problem of optimal boundary control of this system is considered the solvability of the control problem is proved and necessary optimality conditions of first order are derived examples of non singularity of these conditions are given

radiation models for thermal flows at low mach number simplified approximate models for radiation are proposed to study thermal effects in low mach flow in open tunnels the governing equations for fluid dynamics are derived by applying a low mach asymptotic in the compressible navier stokes problem based on an asymptotic analysis we show that the integro differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable and easy to solve using standard numerical discretizations as an application we consider a simplified fire model in vehicular tunnels the results presented in this paper show that the proposed models are able to predict temperature in the tunnels accurately with low computational cost

time independent hybrid enrichment for finite element solution of transient conduction radiation in diffusive grey media we investigate the effectiveness of the partition of unity finite element method for transient conduction radiation problems in diffusive grey media the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media the coupled equations are integrated in time using a semi implicit method in the finite element framework we show that for the considered problems a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method it is illustrated that this approach can be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls the performance of the proposed partition of unity method is analyzed on several test examples for transient conduction radiation problems in two space dimensions

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

solvability of p1 approximation of a conductive radiative heat transfer problem conductive radiative heat transfer in a scattering and absorbing medium bounded by two reflecting and radiating plane surfaces is examined p1 approximation diffusion model is used for the simplification of the original problem the existence and uniqueness of solutions of the diffusion model are proved the convergence of an iterative procedure for finding solutions is proved numerical examples are discussed

modeling radio wave propagation in tunnels with a vectorial parabolic equation to study radio wave propagation in tunnels we present a vectorial parabolic equation pe taking into account the cross section shape wall impedances slowly varying curvature and torsion of the tunnel axis for rectangular cross section two polarizations are decoupled and two families of adiabatic modes can be found explicitly giving a generalization of the known results for a uniform tunnel in the general case a boundary value problem arises to be solved by using finite difference finite element fd fe techniques numerical examples demonstrate the computational efficiency of the proposed method

boundary optimal control problem of complex heat transfer model abstract a boundary optimal control problem of complex heat transfer is under consideration the problem is composed of a nonstationary heat transfer equation and the steady state p 1 approximation of the radiative heat transfer the control parameter is related to the reflection coefficient of the boundary the solvability of the control problem is proven and the optimality system is derived results of numerical simulations are presented

model reduction techniques for frequency averaging in radiative heat transfer we study model reduction techniques for frequency averaging in radiative heat transfer especially we employ proper orthogonal decomposition in combination with the method of snapshots to devise an automated a posteriori algorithm which helps to reduce significantly the dimensionality for further simulations the reliability of the surrogate models is tested and we compare the results with two other reduced models which are given by the approximation using the weighted sum of gray gases and by an frequency averaged version of the so called mathrm sp n model we present several numerical results underlining the feasibility of our approach

a gaussian beam approximation approach for embedding antennas into vector parabolic equation based wireless channel propagation models vector parabolic equation vpe methods have been widely applied to the modeling of radio wave propagation in tunnel environments offering high computational efficiency and fidelity while the propagation environment can be discretized and represented in detail the representation of radiating sources such as transmitting antennas requires the calculation analytical if possible or numerical via another method such as ray tracing rt of the fields that the sources generate on the initial plane of the vpe model these initial conditions are necessary for subsequently applying vpe however the solutions offered so far compromise either the accuracy or the efficiency of vpe for example generating the initial conditions for vpe through rt adds significant computational overhead to the typically fast vpe solver to address this significant limitation of vpe methods we introduce a technique that allows one to directly embed antennas into a vpe mesh via a gaussian beam approximation of their radiated fields hence the initial conditions for vpe are generated for practical antenna patterns without invoking other techniques and with no compromise on the inherent efficiency of vpe concrete guidelines on how to choose parameters for gaussian beams are provided numerical results are compared to experimental measurements in various tunnel scenarios demonstrating the validity and usefulness of the technique

splittings of finitely generated groups over two ended subgroups we describe a means of constructing splittings of a one ended finitely generated group over two ended subgroups starting with a finite collection of codimension one two ended subgroups in the case where all the two ended subgroups have two ended commensurators we obtain an annulus theorem and a form of jsj splitting of rips and sela the construction uses ideas from the work of dunwoody sageev and swenson we use a particular kind of order structure which combines cyclic orders and treelike structures in the special case of hyperbolic groups this provides a link between combinarorial constructions and constructions arising from the topological structure of the boundary in this context we recover the annulus theorem of scott and swarup we also show that a one ended finitely generated groups which contains an infinite order element and such that every infinite cyclic subgroup is virtually codimension one is a virtual surface group

asymptotic derivation and numerical investigation of time dependent simplified p n equations the steady state simplified p n sp n approximations to the linear boltzmann equation have been proven to be asymptotically higher order corrections to the diffusion equation in certain physical systems in this paper we present an asymptotic analysis for the time dependent simplified p n equations up to n 3 additionally sp n equations of arbitrary order are derived in an ad hoc way the resulting sp n equations are hyperbolic and differ from those investigated in a previous work by some of the authors in two space dimensions numerical calculations for the p n and sp n equations are performed we simulate neutron distributions of a moving rod and present results for a benchmark problem known as the checkerboard problem the sp n equations are demonstrated to yield significantly more accurate results than diffusion approximations in addition for sufficiently low values of n they are shown to be more efficient than p n models of comparable cost

an iterative method for solving a complex heat transfer problem conductive radiative heat transfer in a scattering and absorbing medium bounded by two reflecting and radiating plane surfaces is considered a diffusion model for describing heat transfer is examined the theorems of existence and uniqueness of a solution of this problem are proved an iterative procedure for solving radiative conductive heat transfer problem is proposed

target scattering calculations with the parabolic equation method the parabolic equation technique is used to solve the helmholtz equation in the presence of scatterers of arbitrary shape in two and three dimensions the scattered field is computed directly using non homogeneous boundary conditions on the scattering object to represent the incident field effectively this decouples the pe paraxial direction from the direction of incidence for convex objects the whole range of scattering angles can be covered with a small number of narrow angle calculations finite difference implementations involve tridiagonal matrices in two dimensions and more general sparse matrices in three dimensions the resulting codes can be used to solve scattering problems for objects ranging in size from a few wavelengths to hundreds of wavelengths the method has been tested against analytical solutions for soft and rigid circular cylinders in 2d and soft and rigid spheres in 3d showing good agreement at all scattering angles

inverse problem with finite overdetermination for steady state equations of radiative heat exchange abstract an inverse problem for a system of semilinear elliptic equations modeling simultaneously conductive and radiative heat transfer is under consideration the problem consists in finding the right hand side of the heat transfer equation in the form of linear combination of given functionals on the base of prescribed values of these functionals on the solution the solvability of the problem is proven without any smallness assumptions it is shown that the set of solutions is homeomorphic to a finite dimensional compact set and conditions of uniqueness of solutions are found

simplified p n approximations to the equations of radiative heat transfer and applications simplified pn spn approximations to the equations of radiative heat transfer are derived for optically thick diffusive systems and appropriate boundary conditions are formulated the spn equations are derived by an asymptotic analysis while the boundary conditions are obtained from a variational analysis numerical comparisons of the spn equilibrium diffusion and full transport equations are presented to demonstrate the accuracy and efficiency of the spn approximations

efficient numerical methods for radiation in gas turbines numerical methods for radiative heat transfer equations coupled to a temperature equation are considered efficient solution methods and approximate equations for this system are investigated and a comparative numerical study of the different approximations is given the approximate equations considered in this paper include moment methods and diffusive approximations fast iterative solvers for the problem like multilevel methods with suitable preconditioning are considered in detail numerical experiments and comparisons in different space dimensions and for various physical situations are presented

a numerical scheme for the solution of the vector parabolic equation governing the radio wave propagation in straight and curved rectangular tunnels a high frequency field prediction model useful to evaluate the radio wave propagation in realistic tunnels having rectangular cross section is presented the model is based on a hybrid implicit explicit finite difference numerical scheme adopted to solve the vector parabolic equation vpe governing the electromagnetic field propagation in straight and curved tunnels to model the materials forming the tunnel walls mixed dirichlet von neumann boundary conditions are employed in this way all the relevant field propagation processes are properly taken into account to show the feasibility of the proposed numerical method results concerning the spatial distribution of the electromagnetic field and the power flux in straight and curved rectangular tunnels are provided an analysis concerning the convergence properties of the proposed numerical scheme is also reported

partial moment entropy approximation to radiative heat transfer we extend the half moment entropy closure for the radiative heat transfer equations presented in dubroca and klar b dubroca a klar half moment closure for radiative transfer equations j comput phys 180 2002 584 596 and turpault et al r turpault m frank b dubroca a klar multigroup half space moment approximations to the radiative heat transfer equations j comput phys 198 2004 363 371 to multi d to that end we consider a partial moment system with general partitions of the unit sphere closed by an entropy minimization principle we give physical and mathematical reasons for this choice of model and study its properties several numerical examples in different physical regimes are presented

cyclic splittings of finitely presented groups and the canonical jsj decomposition the classification of stable actions of finitely presented groups on trees has found a number of applications perhaps one of the most striking of these applications is the theory of canonical jsj decompositions of hyperbolic groups developed by sela in his seminal paper se

nondegeneracy of optimality conditions in control problems for a radiative conductive heat transfer model a boundary control problem for a nonlinear steady state heat transfer model accounting for heat radiation effects is considered the problem consists in the minimization of a cost functional by controlling the reflection properties of the boundary the solvability of the control problem is proven an optimality system is derived and the nondegeneracy of optimality conditions is established the results of numerical simulations are presented

a linear algebraic analysis of diffusion synthetic acceleration for the boltzmann transport equation a linear algebraic formulation of discretized mono energetic steady state linear boltzmann transport equations in slab geometry is presented the discretization consists of a discrete ordinates collocation in angle and a diamond difference method in space by expressing diffusion synthetic acceleration in this formalism asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various asymptotic regimes including the asymptotic diffusion limit these results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux and or reflection prescribed at the boundaries numerical results are also presented which demonstrate these results

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

error estimates for the combined spatial and angular approximations of the transport equation for slab geometry we study the convergence of the discrete ordinates method in the numerical solution of the transport equation for slab geometry in particular the combined effect of spatial and angular approximations is analyzed general conditions are given which guarantee the stability of the combined approximations error estimates are derived for both the scalar flux in the l p norm 1 leqq p leqq infty and for the approximate critical eigenvalues a comparison with numerical results indicates that the eigenvalue error estimates are optimal

the unique solvability of a complex 3d heat transfer problem abstract conductive convective radiative heat transfer in a scattering and absorbing medium with reflecting and radiating boundaries is considered the p 1 approximation diffusion model is used for the simplification of the original problem the existence of bounded states of the diffusion model is proved the uniqueness of solutions is established under certain assumptions

a fast multigrid algorithm for isotropic transport problems i pure scattering the authors present a multigrid method for solving the one dimensional 1 d slab geometry s n equations with isotropic scattering and no absorption this scheme is highly compatible with massively parallel computer architectures and represents a first step toward similar multigrid methods for the s n equations in curvilinear and multidimensional geometries extensive theoretical analyses are given for our scheme which indicate that it is extremely efficient in fact the method is so efficient that it very nearly represents an exact solution technique results from calculations are presented which validate the theoretical results the case with absorption is treated in a sequel to this paper manteuffel et al a fast multigrid algorithm for transport problems ii with absorption siam j sci comput submitted

an enriched finite element model with q refinement for radiative boundary layers in glass cooling radiative cooling in glass manufacturing is simulated using the partition of unity finite element method the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary simplified p 1 approximation for the radiation in non grey semitransparent media to integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework a class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls using an industrial electromagnetic spectrum the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h version finite element method furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries the time integration choice is essential to implement a q refinement procedure introduced in the current study the enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment the performance of the proposed method is analysed for glass annealing in two enclosures where the simplified p 1 approximation solution with the partition of unity method the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical rosseland model

numerical solution of nonlinear inverse heat conduction problem with a radiation boundary condition a finite difference solution for the transient nonlinear heat conduction equation in a finite slab with a radiation boundary condition is proposed an implicit finite difference approximation is used which enables accurate estimation of the surface temperature as well as prevention of oscillation of computed values at the surfaces a two time level implicit method is used while taylor s forward projection method is employed for taking account of the nonlinearities an iterative method is described which predicts unknown surface conditions an example is illustrated which is typical of those that arise in practical applications the results demonstrate that the method is remarkable in its stability to predict surface conditions without debilitation

the algebraic torus theorem a structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup

optimal boundary control of a steady state heat transfer model accounting for radiative effects abstract a boundary control problem for a nonlinear steady state heat transfer model accounting for heat radiation effects is considered the aim of control consists in obtaining a prescribed temperature distribution in a part of the model domain by controlling the boundary temperature the solvability of this control problem is proven and optimality conditions are derived numerical simulations are presented

a fast multilevel algorithm for the solution of nonlinear systems of conductive radiative heat transfer equations in two space dimensions in this paper we describe a fast multilevel algorithm for the solution of a system of nonlinear integro differential equations that model steady state combined conductive radiative heat transfer in two space dimensions this extends our previous work in one space dimension we formulate the equations as a compact fixed point problem with the temperature as the unknown the fixed point map requires both a poisson solve and a transport solve for its evaluation as a solver for both the transport problem and the full system we apply the atkinson brakhage algorithm using newton gmres as the solver on the coarse mesh we compare our solver choices with newton gmres under modest stability and convergence assumptions on the transport solver we prove convergence of the multilevel method for the complete system

half moment closure for radiative transfer equations in this paper a moment method for radiative transfer equations is considered which has been developed and investigated using different approaches problems appearing for this moment system for boundary value problems using maxwell type boundary conditions are described a new method based on the consideration of positive and negative half fluxes is developed and shown to overcome the above problems moreover a numerical scheme and numerical results for the new moment system are presented

an inverse problem for a quasi static approximate model of radiative heat transfer abstract we consider an inverse problem for a system of equations modeling quasi static conductive and radiative heat transfer the problem consists in finding the right hand side of the heat transfer equation which is a linear combination of given functionals the prescribed data are the values of these functionals evaluated on the solution the solvability of the problem is proven without any smallness assumptions on the model parameters in the class of bounded temperature fields the uniqueness of the solution of the inverse problem is shown further we study the tikhonov regularization in the framework of a pde constrained optimization problem and show that the approximating sequence contains a convergent subsequence the analytical results depend crucially on new and refined a priori estimates for the solutions

new frequency averaged approximations to the equations of radiative heat transfer we develop new direction and frequency averaged approximations to the equations of radiative heat transfer in glass for optically thick diffusive regimes these approximations which are based on the rm sp n approach given in e larsen g thommes a klar m seaid and t gotz j comput phys 83 2002 pp 652 675 represent asymptotic corrections to the familiar rosseland or equilibrium diffusion approximation numerical results for realistic problems in the simulation of radiative heat transfer in glass cooling confirm the accuracy and efficiency of the new approximations

theoretical and numerical analysis of a minimal residual solver for 2d boltzmann transport equation relying on the splitting of the collision operator introduced in m2an 34 2000 1109 m2an 32 1998 341 we prove theoretical convergence for an infinite dimensional adaptation of the minimal residual algorithm for boltzmann transport equation in dimension two then we compare numerically this method with existing nonaccelerated schemes it gives good results which could even be further improved by adding a dsa or sor acceleration

hp adaptive discontinuous galerkin methods for simplified pn approximations of frequency dependent radiative transfer abstract we investigate the performance of a class of h p adaptive discontinuous galerkin methods for the numerical solution of simplified p n approximations of radiative transfer in non gray semitransparent media by introducing an optical scale and using asymptotic expansions in the radiative transfer equation we formulate the simplified p n approximations the optical spectrum is decomposed in frequency bands and the simplified p n equations are solved for each frequency band as a numerical solver for the simplified p n equations we consider a high order discontinuous galerkin method the solver belongs to a class of finite element methods whose approximate solutions are discontinuous across inter element boundaries this property renders the method ideally suited for the h p adaptivity an error estimator is shown to provide reliable and practically useful upper bounds for the numerical errors independent of the optical scales used in the simulations the proposed method is simple fast and highly accurate the performance of the method is analyzed on several applications in frequency dependent radiative transfer the aim of such a method compared to the conventional finite element methods is to solve the simplified p n equations efficiently and with a high level of accuracy on unstructured meshes with different elements the obtained results demonstrate the ability of the proposed method to capture the main radiative features

analogues of cayley graphs for topological groups we define for a compactly generated totally disconnected locally compact group a graph called a rough cayley graph that is a quasi isometry invariant of the group this graph carries information about the group structure in an analogous way to the ordinary cayley graph for a finitely generated group with this construction the machinery of geometric group theory can be applied to topological groups this is illustrated by a study of groups where the rough cayley graph has more than one end and a study of groups where the rough cayley graph has polynomial growth

a finite difference solution of non linear systems of radiative conductive heat transfer equations a finite difference solution for a system of non linear integro differential equations modelling the steady state combined radiative conductive heat transfer is proposed a new backward forward finite difference scheme is formulated for the radiative transfer equation the non linear heat conduction equation is solved using the kirchhoff transformation associated with a centred finite difference scheme the coupled system of equations is solved using a fixed point method which relates to the temperature field an application on a real insulator composed of silica fibres is illustrated the results show that the method is very efficient copyright 2002 john wiley sons ltd

exact surface impedance admittance boundary conditions for complex geometries theory and applications a methodology useful to derive exact and higher order surface impedance admittance boundary conditions hoi abc s for complex geometries is presented it is shown that exact surface boundary conditions are always expressed through dyadic integral operators involving the tangential magnetic and electric fields all over the surface of the body quasi local surface boundary conditions that include curvature effects are shown to be computable through an asymptotic approximation of the integral operators finally an example of a surface admittance boundary condition useful to analyze a structure exhibiting discontinuities along its surface boundary is presented practical examples to demonstrate the feasibility of the proposed methodology as well as the accuracy of the resulting surface boundary conditions are also presented

finite difference approximations and superconvergence for the discrete ordinate equations in slab geometry a unified framework is developed for calculating the order of the error for a class of finite difference approximations to the monoenergetic linear transport equation in slab geometry in particular the global discretization errors for the step characteristic diamond and linear discontinuous methods are shown to be of order two while those for the linear moments and linear characteristic methods are of order three and that for the quadratic method is of order four a superconvergence result is obtained for the three linear methods in the sense that the cell averaged flux approximations are shown to converge at one order higher than the global errors

multigrid newton krylov method for radiation in diffusive semitransparent media we present a fast multigrid solver for simplified p n sp n approximations to the diffusive radiation in non grey semitransparent media the method consists on reformulating the equations as a nonlinear fixed point problem in the temperature only given a mesh hierarchy time and space discretizations are performed using second order implicit and finite differencing methods respectively at each mesh level a newton krylov algorithm is applied to the discrete equations as a smoother on the coarse meshes we propose the atkinson brakhage operator numerical results are shown for glass cooling process using different geometry enclosures the sp n approximations capture the correct asymptotic behavior of the numerical solution with a computational cost lower than using the full radiative transfer equations