discretization of the multiscale semiconductor boltzmann equation by diffusive relaxation schemes in this paper we derive diffusive relaxation schemes for the linear semiconductor boltzmann equation that work in both the kinetic and diffusive regimes similar to our earlier approach for multiscale transport equations we use the even and odd parity formulation of the kinetic equation and then reformulate it into the diffusive relaxation system drs in order to handle the implicit anisotropic collision term efficiently we utilize a suitable power series expansion based on the wild sum which yields a time discretization uniformly stable with any desired order of accuracy yet is explicitly solvable with the correct drift diffusion limit the velocity discretization is done with the gauss hermite quadrature rule equivalent to a moment expansion method asymptotic analysis and numerical experiments show that the schemes have the usual advantages of a diffusive relaxation scheme for multiscale transport equations and are asymptotic preserving

consensus of high order multi agent systems with large input and communication delays we study in this paper the consensus problem for multi agent systems with agents characterized by high order linear systems with time delays in both the communication network and inputs provided that the open loop dynamics of the agents is not exponentially unstable but may be polynomially unstable and the communication topology contains a directed spanning tree a truncated predictor feedback approach is established to solve the consensus problem it is shown that if the delays are constant and exactly known the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays if it is further assumed that the open loop dynamics of the agents only contains zero eigenvalues the delays are allowed to be time varying and unknown numerical examples are worked out to illustrate the effectiveness of the proposed approaches

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

validation of the shnc time dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies abstract the diffusion approximation to the time dependent boltzmann transport equation gives accurate results for traditional nuclear reactor designs but new reactor designs and new fuel elements require neutron transport methods we develop a numerical approximation to the time dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics p l equations for odd l and on the backward euler finite difference discretization of time the resulting scheme can be written as a stationary form of diffusive second order p l equations this allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem using coarse spatial grids to further reduce memory requirements this scheme is validated with several transient benchmarks where the convergence properties are established and compared with the simplified p l approximation a more realistic transient benchmark based on the two group c5 mox problem is finally introduced showing the need of high order p l approximation for complex fuel geometries

stability of the supg finite element method for transient advection diffusion problems implicit time integration coupled with supg discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows recently it has been suggested that for small courant numbers these terms may dominate the streamline diffusion term ostensibly causing destabilization of the supg method while consistent with a straightforward finite element stability analysis this contention is not supported by computational experiments and contradicts earlier von neumann stability analyses of the semidiscrete supg equations this prompts us to re examine finite element stability of the fully discrete supg equations a careful analysis of the additional terms reveals that regardless of the time step size they are always dominated by the consistent mass matrix consequently supg cannot be destabilized for small courant numbers numerical results that illustrate our conclusions are reported

a comparison of implicit time integration methods for nonlinear relaxation and diffusion several time integration methods for nonlinear systems are compared all of the time discretizations are based on the method but differ in their treatment of the implicit nonlinear terms one method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent nc newton s method or its approximation newton krylov is used to converge the nonlinearities the other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem the linearized one step method that uses the full jacobian is shown to have similar accuracy as nc methods the lagged linearization method and an extension that is second order accurate are also studied a trunction error analysis complements the numerical results for the relaxation problem it is shown that each of the second order accurate linearized methods may be more accurate than an nc method depending on the degree of nonlinearity in the problem for the radiation diffusion problem in general the nc method is most accurate and allows a larger time step however the linearized methods perfom surprisingly well

an asymptotic preserving two dimensional staggered grid method for multiscale transport equations we propose a two dimensional asymptotic preserving scheme for linear transport equations with diffusive scalings it is an extension of the time splitting developed by jin pareschi and toscani siam j numer anal 38 2000 pp 913 936 but uses spatial discretizations on staggered grids which preserves the discrete diffusion limit with a more compact stencil the first novelty of this paper is that we propose a staggering in two dimensions that requires fewer unknowns than one could have naively expected the second contribution of this paper is that we rigorously analyze the scheme of jin pareschi and toscani we show that the scheme is ap and obtain an explicit cfl condition which couples a hyperbolic and a parabolic condition this type of condition is common for asymptotic preserving schemes and guarantees uniform stability with respect to the mean free path in addition we obtain an upper bound on the relaxation parameter which is the crucial parameter of the used time discretization se

radiation transport modeling using extended quadrature method of moments abstract the radiative transfer equation describes the propagation of radiation through a material medium while it provides a highly accurate description of the radiation field the large phase space on which the equation is defined makes it numerically challenging as a consequence significant effort has gone into the development of accurate approximation methods recently an extended quadrature method of moments eqmom has been developed to solve univariate population balance equations which also have a large phase space and thus face similar computational challenges the distinct advantage of the eqmom approach over other moment methods is that it generates moment equations that are consistent with a positive phase space density and has a moment inversion algorithm that is fast and efficient the goal of the current paper is to present the eqmom method in the context of radiation transport to discuss advantages and disadvantages and to demonstrate its performance on a set of standard one dimensional benchmark problems that encompass optically thin thick and transition regimes special attention is given in the implementation to the issue of realizability that is consistency with a positive phase space density numerical results in one dimension are promising and lay the foundation for extending the same framework to multiple dimensions

remarks on unboundedness of set valued ito stochastic integrals abstract the paper deals with set valued stochastic integrals which are defined as a random multifunctions integrable boundedness of a such integral is one of the most important features in potential applications unfortunately up to now there were no correct proofs of such property surprisingly also a negative answer to this problem is still not explained correctly hence the problem seems to be still undetermined we shall show that in general the answer is negative we shall provide several relatively simple examples both in convex and nonconvex valued case as a consequence we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set valued ito s integral is equivalent to its single valuedness

asymptotic induced domain decomposition methods for kinetic and drift diffusion semiconductor equations this paper deals with domain decomposition methods for kinetic and drift diffusion semiconductor equations in particular accurate coupling conditions at the interface between the kinetic and drift diffusion domain are given the cases of slight and strong nonequilibrium situations at the interface are considered and numerical examples are shown

efficient solution of the simplified p n equations in this paper we show new solver strategies for the multigroup sp n equations for nuclear reactor analysis by forming the complete matrix over space moments and energy a robust set of solution strategies may be applied power iteration shifted power iteration rayleigh quotient iteration arnoldi s method and a generalized davidson method each using algebraic and physics based multigrid preconditioners have been compared on the c5g7 mox test problem as well as an operational pressurized water reactor model our results show that the most efficient approach is the generalized davidson method which is 30 40 times faster than traditional power iteration and 6 10 times faster than arnoldi s method

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

consensus reaching in multi agent packet switched networks with non linear coupling an important task for multi agent systems mas is to reach a consensus e g to align their velocity vectors recent results propose appropriate consensus protocols to achieve such tasks but most of them do not consider the effect of communication constraints such as the presence of time delays in the exchange of information between the agents in this article we provide conditions for a non linear locally passive mas of arbitrary size to reach a consensus when the agents communicate over a packet switched network that is characterised by a given topology both the cases of constant and switching topologies are considered the nature of the communication channel imposes constraints that are modelled using stochastic delays of arbitrary distribution we first embed this model in another distributed but deterministic delay model and provide conditions for the error introduced by this simplification in our main result we provide conditions for the locally passive mas with distributed delays to reach

locally lipschitz selections in banach lattices abstract let x be a banach space while y is a banach lattice we consider the class of upper separated set valued functions f x 2 y and investigate the problem of the existence of convex and locally lipschitz selections of f we discuss some applications of the selection results obtained in the paper to the theory of deterministic and stochastic differential inclusions

uniformly accurate diffusive relaxation schemes for multiscale transport equations many transport equations such as the neutron transport radiative transfer and transport equations for waves in random media have a diffusive scaling that leads to the diffusion equations in many physical applications the scaling parameter mean free path may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic diffusive regimes within one problem and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter in an earlier work jin pareschi and toscani siam j numer anal 35 1998 pp 2405 2439 we handled this numerical problem for discrete velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws jin and xin comm pure appl math 48 1995 pp 235 277 such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit yet explicitly implementable schemes that work with high resolution uniformly with respect to the relaxation parameter in this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities when one uses the even and odd parities of the transport equation

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

stochastic consentability of linear systems with time delays and multiplicative noises this paper develops stochastic consentability of linear multiagent systems with time delays and multiplicative noises first the stochastic stability for stochastic differential delay equations driven by multiplicative noises is examined and the existence of the positive definite solution for a class of generalized algebraic riccati equations gares is established then sufficient conditions are deduced for the mean square and almost sure consentability and stabilization based on the developed stochastic stability and gares consensus protocols are designed for linear multiagent systems with undirected and leader following topologies it is revealed that multiagent consentability depends on certain characterizing system parameters including linear system dynamics communication graph channel uncertainties and time delay of the deterministic term it is shown that a second order integrator multiagent system is unconditionally mean square and almost surely consentable for any given noise intensities and time delay and that the mean square and almost sure consensus can be achieved by carefully choosing the control gain according to certain explicit conditions

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

a low rank method for two dimensional time dependent radiation transport calculations abstract the low rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank which has been applied to the simulation of high dimensional problems to reduce the memory required and computational cost in this work a dynamical low rank approximation method is developed for the time dependent radiation transport equation in 1 d and 2 d cartesian geometries using a finite volume discretization in space and a spherical harmonics basis in angle we construct a system that evolves on a low rank manifold via an operator splitting approach numerical results on five test problems demonstrate that the low rank solution requires less memory and computational time than solving the full rank equations with the same accuracy it is furthermore shown that the low rank algorithm can obtain high fidelity results by increasing the number of basis functions while keeping the rank fixed

mas consensus and delay limits under delayed output feedback in this technical note we study the consensus problem for discrete time multi agent systems over an undirected fixed network communication graph focusing on the robustness of consensus with respect to communication delay we assume that the agents input is subject to a constant albeit possibly unknown time delay and is generated by a distributed dynamic output feedback control protocol drawing upon concepts and techniques from robust control notably those concerning gain margin optimization and analytic interpolation we derive explicit closed form conditions for general linear agents to achieve consensus our results display an explicit dependence of the consensus condition on the delay value as well as on the agent s unstable poles and non minimum phase zeros showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology we also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method

multi species neutron transport equation the neutron transport equation nte describes the flux of neutrons through inhomogeneous fissile medium whilst well treated in the nuclear physics literature the nte has had a somewhat scattered treatment in mathematical literature with a variety of different approaches within a probabilistic framework it has somewhat undeservingly received little attention in recent years nonetheless a few probabilistic treatments can be found in this article our aim is threefold first we want to introduce a slightly more general setting for the nte which gives a more complete picture of the different species of particle and radioactive fluxes that are involved in fission second we consolidate the classical c 0 semigroup approach to solving the nte with the method of stochastic representation which involves expectation semigroups third we provide the leading asymptotic of our multi species nte which will turn out to be crucial for further stochastic analysis of the nte in forthcoming work cox et al 2019 harris et al 2018 horton et al 2018 the methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c 0 semigroup theory from functional analysis in this respect our presentation is thus part review of existing theory and part presentation of new research results based on generalisation of existing results

second order discretization in space and time for radiation hydrodynamics second order accurate discretizations for radiation hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities second order methods used to solve the hydrodynamics equations and second order methods used to solve the radiation transport equation often differ fundamentally making it difficult to combine them in a second order manner here we present an implicitexplicit imex method for solving the 1d equations of radiation hydrodynamics that is second order accurate in space and time our radiation hydrodynamics model consists of the 1d euler equations coupled with a gray radiation s2 approximation our rh method combines the muscl hancock method for solving the euler equations with the tr bdf2 time integration scheme and the linear discontinuous galerkin finite element spatial discretization scheme for the s2 radiation equations the muscl hancock method is commonly used for hydrodynamic calculations and the linear discontinuous galerkin scheme is the standard for the sn equations of radiative transfer while somewhat similar these schemes vary fundamentally with respect to the treatment of spatial slopes we address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome using the method of manufactured solutions we show that the method is second order accurate in space and time for both the equilibrium diffusion and streaming limit and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi analytic solutions

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

spatial discretizations for self adjoint forms of the radiative transfer equations there are three commonly recognized second order self adjoint forms of the neutron transport equation the even parity equations the odd parity equations and the self adjoint angular flux equations because all of these equations contain second order spatial derivatives and are self adjoint for the mono energetic case standard continuous finite element discretization techniques have proved quite effective when applied to the spatial variables we first derive analogs of these equations for the case of time dependent radiative transfer the primary unknowns for these equations are functions of the angular intensity rather than the angular flux hence the analog of the self adjoint angular flux equation is referred to as the self adjoint angular intensity equation then we describe a general arbitrary order continuous spatial finite element approach that is applied to each of the three equations in conjunction with backward euler differencing in time we refer to it as the standard technique we also introduce an alternative spatial discretization scheme for the self adjoint angular intensity equation that requires far fewer unknowns than the standard method but appears to give comparable accuracy computational results are given that demonstrate the validity of both of these discretization schemes

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

on the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models the unified gas kinetic scheme ugks of k xu et al 2010 37 originally developed for multiscale gas dynamics problems is applied in this paper to a linear kinetic model of radiative transfer theory while such problems exhibit purely diffusive behavior in the optically thick or small knudsen regime we prove that ugks is still asymptotic preserving ap in this regime but for the free transport regime as well moreover this scheme is modified to include a time implicit discretization of the limit diffusion equation and to correctly capture the solution in case of boundary layers contrary to many ap schemes this method is based on a standard finite volume approach it does neither use any decomposition of the solution nor staggered grids several numerical tests demonstrate the properties of the scheme

specular reflection treatment for the 3d radiative transfer equation solved with the discrete ordinates method the contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems the radiative transfer equation being integro differential the discrete ordinates method allows to write down a set of semi discrete equations in which weights are to be calculated the calculation of these weights is well known to be based on either a quadrature or on angular discretization making the use of such method straightforward for the state equation also the diffuse contribution of reflection on borders is usually well taken into account however the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders this paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments the developed algorithms combined with a decentered finite element scheme are validated with the help of comparisons with analytical solutions before being applied on complex geometries

linear multifrequency grey acceleration recast for preconditioned krylov iterations the linear multifrequency grey acceleration lmfga technique is used to accelerate the iterative convergence of multigroup thermal radiation diffusion calculations in high energy density simulations although it is effective and efficient in one dimensional calculations the lmfga method has recently been observed to significantly degrade under certain conditions in multidimensional calculations with large discontinuities in material properties to address this deficiency we recast the lmfga method in terms of a preconditioned system that is solved with a krylov method lmfgk results are presented demonstrating that the new lmfgk method always requires fewer iterations than the original lmfga method the reduction in iteration count increases with both the size of the time step and the inhomogeneity of the problem however for reasons later explained the lmfgk method can cost more per iteration than the lmfga method resulting in lower but comparable efficiency in problems with small time steps and weak inhomogeneities in problems with large time steps and strong inhomogeneities the lmfgk method is significantly more efficient than the lmfga method

symbolic implicit monte carlo radiation transport in the difference formulation a piecewise constant discretization the equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit one source of this difficulty is that in thick absorbing media thermal emission and absorption are almost completely balanced a new formulation for thermal radiation transport called the difference formulation was recently introduced in order to remove the stiff balance between emission and absorption in the new formulation thermal emission is replaced by derivative terms that become small in thick media it was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation while preserving full rigor and accuracy of the transport solution in the streaming limit in this paper the transport equation is solved by the symbolic implicit monte carlo method and comparisons are made between the standard formulation and the difference formulation the method is easily adapted to the derivative source terms of the difference formulation and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media

least squares finite element solution of the neutron transport equation in diffusive regimes a systematic solution approach for the neutron transport equation based on a least squares finite element discretization is presented this approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution bounds for the discretization error and guidance for the development of an efficient multigrid solver for the resulting discrete problem to guarantee the accuracy of the discrete solution for diffusive regimes a scaling transformation is applied to the transport operator prior to the discretization the key result is the proof of the v ellipticity and continuity of the scaled least squares bilinear form with constants that are independent of the total cross section and the absorption cross section for a variety of least squares finite element discretizations this leads to error bounds that remain valid in diffusive regimes moreover for problems in slab geometry a full multigrid solver is presented with v 1 1 cycle convergence factors approximately equal to 0 1 independent of the size of the total cross section and the absorption cross section

a second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media a new second order form of radiative transfer equation named msorte is proposed which overcomes the singularity problem of a previously proposed second order radiative transfer equation j e morel b t adams t noh j m mcghee t m evans t j urbatsch spatial discretizations for self adjoint forms of the radiative transfer equations j comput phys 214 1 2006 12 40 where it was termed saai j m zhao l h liu second order radiative transfer equation and its properties of numerical solution using finite element method numer heat transfer b 51 2007 391 409 in dealing with inhomogeneous media where some locations have very small zero extinction coefficient the msorte contains a naturally introduced diffusion or second order term which provides better numerical property than the classic first order radiative transfer equation rte the stability and convergence characteristics of the msorte discretized by central difference scheme is analyzed theoretically and the better numerical stability of the second order form radiative transfer equations than the rte when discretized by the central difference type method is proved a collocation meshless method is developed based on the msorte to solve radiative transfer in inhomogeneous media several critical test cases are taken to verify the performance of the presented method the collocation meshless method based on the msorte is demonstrated to be capable of stably and accurately solve radiative transfer in strongly inhomogeneous media media with void region and even with discontinuous extinction coefficient

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport abstract this paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes for a linear discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns in a process known as source tilting the asymptotic limits of both monte carlo continuous in space and deterministic approaches based on linear discontinuous finite elements for solving the transport equation are investigated in slab geometry the resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell edge discontinuity in the temperature representation based on this analysis it is possible to design accurate schemes for representing the material temperature for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ale arbitrary lagrange eulerian hydrodynamics schemes

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

analysis of a numerical solver for radiative transport equation we analyze a numerical algorithm for solving radiative transport equation with vacuum or reflection boundary condition that was proposed in 4 with angular discretization by finite element method and spatial discretization by discontinuous galerkin or finite difference method

average consensus in multi agent systems with uncertain topologies and multiple time varying delays abstract in this paper we investigate average consensus problem in continuous time multi agent systems with uncertain topologies as well as multiple time varying communication delays the network of dynamic agents is directed and both fixed and switching topologies are considered by using the linear matrix inequality method we prove that all the nodes in the network can achieve average consensus asymptotically for appropriate delays and uncertainties if the network topology is weakly connected and balanced feasible linear matrix inequalities are established to determine the maximal allowable upper bounds of multiple delays numerical examples are presented to illustrate the availability of the theoretical results

semi implicit time integration for pn thermal radiative transfer implicit time integration involving the solution of large systems of equations is the current paradigm for time dependent radiative transfer in this paper we present a semi implicit linear discontinuous galerkin method for the spherical harmonics p n equations for thermal radiative transfer in planar geometry our method is novel in that the material coupling terms are treated implicitly via linearizing the emission source and the streaming operator is treated explicitly using a second order accurate runge kutta method the benefit of this approach is that each time step only involves the solution of equations that are local to each cell this benefit comes at the cost of having the time step limited by a cfl condition based on the speed of light to guarantee positivity and avoid artificial oscillations we use a slope limiting technique we present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean free path is not resolved by the spatial mesh also in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion cfl condition we demonstrate with numerical results that away from the diffusion limit our method demonstrates second order error convergence as the spatial mesh is refined with a fixed cfl number

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

on a class of uniformly accurate imex runge kutta schemes and applications to hyperbolic systems with relaxation in this paper we consider hyperbolic systems with relaxation in which the relaxation time varepsilon may vary from values of order one to very small values when varepsilon is very small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results in such cases it is important to have schemes that work uniformly with respect to varepsilon implicit explicit imex runge kutta r k schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time we develop new imex r k schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit i e when varepsilon is very small these schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time we propose the construction of new third order imex r k schemes of type ck s boscarino siam j numer anal 45 2008 pp 1600 1621 in several test problems these schemes with a fixed spatial discretization exhibit for all range of the relaxation time an almost uniform third order accuracy

a homotopy perturbation method for solving a neutron transport equation the purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method hpm the results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

mesh independence of the generalized davidson algorithm abstract we give conditions under which the generalized davidson algorithm for eigenvalue computations is mesh independent in this case mesh independence means that the iteration statistics residual norms convergence rates for example of a sequence of discretizations of a problem in a banach space converge the statistics for the infinite dimensional problem we illustrate the result with several numerical examples

frequency domain optical tomography based on the equation of radiative transfer optical tomography consists of reconstructing the spatial distribution of absorption and scattering properties of a medium from surface measurements of transmitted light intensities mathematically this problem amounts to parameter identification for the equation of radiative transfer ert with diffusion type boundary measurements because they are posed in the phase space radiative transfer equations are quite challenging to solve computationally most past works have considered the steady state ert or the diffusion approximation of the ert in both cases substantial cross talk has been observed in the reconstruction of the absorption and scattering properties of inclusions in this paper we present an optical tomographic reconstruction algorithm based on the frequency domain ert the inverse problem is formulated as a regularized least squares minimization problem in which the mismatch between forward model predictions and measurements is minimized the ert is discretized by using a discrete ordinates method for the directional variables and a finite volume method for the spatial variables a limited memory quasi newton algorithm is used to minimize the least squares functional numerical simulations with synthetic data show that the cross talk between the two optical parameters is significantly reduced in reconstructions based on frequency domain data as compared to those based on steady state data

uniform stability of a finite difference scheme for transport equations in diffusive regimes an asymptotic preserving numerical scheme with respect to diffusion scalings for a linear transport equation is investigated the scheme is adopted from a class of schemes developed in s jin l pareschi and g toscani siam j numer anal 38 2000 pp 913 936 and a klar siam j numer anal 35 1998 pp 1073 1094 stability is proven uniformly in the mean free path under a cfl type condition turning into a parabolic cfl condition in the diffusion limit

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

continuous time average consensus under dynamically changing topologies and multiple time varying delays abstract the average consensus in continuous time multi agent systems with dynamically changing topologies and multiple time varying delays is studied in this paper the network topology is captured by weighted digraphs which are weakly connected and balanced some feasible linear matrix inequalities are established to determine the allowable upper bounds of multiple delays that guarantee the average consensus of the system numerical examples are provided to show the usefulness of the theoretical results

a monte carlo synthetic acceleration method for solving the thermal radiation diffusion equation we present a novel synthetic acceleration based monte carlo method for solving the equilibrium thermal radiation diffusion equation in three spatial dimensions the algorithm performance is compared against traditional solution techniques using a marshak benchmark problem and a more complex multiple material problem our results show that our monte carlo method is an effective solver for sparse matrix systems for solutions converged to the same tolerance it performs competitively with deterministic methods including preconditioned conjugate gradient and gmres we also discuss various aspects of preconditioning the method and its general applicability to broader classes of problems

delay robustness in non identical multi agent systems we investigate the robustness of consensus protocols on sensor networks to different types of feedback delays we provide set valued conditions for consensus of linear multi agent systems mas with nonidentical agent dynamics and heterogeneous delays these conditions consider different feedback delay configurations and are robust and scalable to unknown arbitrary large topologies and unknown but bounded self delays and communication delays moreover we provide consensus conditions for relative degree two mas with different feedback delays in order to illustrate the applicability of these set valued conditions

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

iterative methods for neutron transport eigenvalue problems we discuss iterative methods for computing criticality in nuclear reactors in general this requires the solution of a generalized eigenvalue problem for an unsymmetric integro differential operator in six independent variables modeling transport scattering and fission where the dependent variable is the neutron angular flux in engineering practice this problem is often solved iteratively using some variant of the inverse power method because of the high dimension matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix free inner iterations this leads to technically complicated inexact iterative methods for which there appears to be no published rigorous convergence theory for the monoenergetic homogeneous model case with isotropic scattering and vacuum boundary conditions we show that before discretization the general nonsymmetric eigenproblem for the angular flux is equivalent to a certain related eigenproblem for the scalar flux involving a symmetric positive definite weakly singular integral operator in space only this correspondence to a symmetric problem in a space of reduced dimension permits us to give a convergence theory for inexact inverse iteration and related methods in particular this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge we also give examples of discretizations which have a corresponding symmetric finite dimensional reduced form the theory is illustrated with numerical examples for several test problems of physical relevance using gmres as the inner solver

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

on the solution of stochastic differential inclusion abstract in the paper the problem of existence and uniqueness of the solution of the stochastic differential inclusion dx t f t x db t g t x dt is investigated under some assumption of the dissipativity type on f and g

a deterministic photon free method to solve radiation transfer equations a new method to solve radiation transfer equations is presented in the absence of scattering material motion and heat conduction the photon variables can be eliminated from the fully implicit multi group discrete ordinate finite difference finite element equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature the solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived the reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission we solve the reduced set of equations by the newton gmres method in which the jacobian update is preconditioned by a linearization of this nonlinear heat equation the performances of this new method and of the semi implicit linear method which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration are compared on two test problems the test results indicate that the new method can take larger time steps requires less memory is more accurate and is competitive in speed with the semi implicit linear method

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

optical tomography reconstruction algorithm with the finite element method an optimal approach with regularization tools optical tomography is mathematically treated as a non linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors according to the ill posed behavior of the inverse problem some regularization tools must be performed and the tikhonov penalization type is the most commonly used in optical tomography applications this paper introduces an optimized approach for optical tomography reconstruction with the finite element method an integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations continuous and discontinuous through a gradient based algorithm where the adjoint method is used to compute the gradient of the cost function an alternative inner product is employed for preconditioning the reconstruction algorithm moreover appropriate re parameterization of the optical properties is performed these regularization strategies are compared with the classical tikhonov penalization one it is shown that both the re parameterization and the use of the sobolev cost function gradient are efficient for solving such an ill posed inverse problem

a numerical method for computing asymptotic states and outgoing distributions for kinetic linear half space problems linear half space problems can be used to solve domain decomposition problems between boltzmann and aerodynamic equations a new fast numerical method computing the asymptotic states and outgoing distributions for a linearized bgk half space problem is presented relations with the so called variational methods are discussed in particular we stress the connection between these methods and chapman enskog type expansions

a wavelet multi scale method for the inverse problem of diffuse optical tomography this paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium this is the so called diffuse optical tomography problem such a non linear ill posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy in a least square sense between some measurements and associated predictions in the present case predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the l bfgs algorithm to cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function a wavelet multi scale method associated with the l bfgs method is developed implemented and validated this method relies on a reformulation of the original inverse problem into a sequence of sub inverse problems of different scales using wavelet transform from the largest scale to the smallest one numerical results show that the proposed method brings more stability with respect to the ordinary l bfgs method and enhances the reconstructed images for most of initial guesses of optical properties

stochastic ito inclusion with upper separated multifunctions abstract we prove the existence of strong solutions of a nonautonomous stochastic inclusion d x t f t x t d t g t x t d w t x 0 with caratheodory upper separated set valued functions f and g

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

asymptotic diffusion limit of the symbolic monte carlo method for the transport equation we use asymptotic analysis to study the diffusion limit of the symbolic implicit monte carlo simc method for the transport equation for standard simc with piecewise constant basis functions we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution we present numerical examples which illustrate the analysis

a residual monte carlo method for discrete thermal radiative diffusion residual monte carlo methods reduce statistical error at a rate of exp bn where b is a positive constant and n is the number of particle histories contrast this convergence rate with 1 n which is the rate of statistical error reduction for conventional monte carlo methods thus residual monte carlo methods hold great promise for increased efficiency relative to conventional monte carlo methods previous research has shown that the application of residual monte carlo methods to the solution of continuum equations such as the radiation transport equation is problematic for all but the simplest of cases however the residual method readily applies to discrete systems as long as those systems are monotone i e they produce positive solutions given positive sources we develop a residual monte carlo method for solving a discrete id non linear thermal radiative equilibrium diffusion equation and we compare its performance with that of the discrete conventional monte carlo method upon which it is based we find that the residual method provides efficiency gains of many orders of magnitude part of the residual gain is due to the fact that we begin each timestep with an initial guess equal to the solution from the previous timestep moreover fully consistent non linear solutions can be obtained in a reasonable amount of time because of the effective lack of statistical noise we conclude that the residual approach has great potential and that further research into such methods should be pursued for more general discrete and continuum systems

on solutions to the pn equations for thermal radiative transfer we present results for the spherical harmonics p n method for solving problems of time dependent thermal radiative transport we prove a theorem that demonstrates that in the streaming limit the spatially and temporally continuous p n equations will allow negative energy densities for any finite order of n we also develop an implicit numerical method for solving the p n equations to explore the impact of the theorem the numerical method uses a high resolution riemann solver to produce an upwinded discretization we employ a quasi linear approach to integrate the nonlinearites added to make the scheme non oscillatory we use the backward euler method for time integration and treat the material interaction terms fully nonlinearly reflecting boundary conditions for the p n equations are presented and we show how to implement this boundary condition using ghost cells the implicit method was able to produce robust results to thermal transport problems in one and two dimensions the numerical method is used to analyze the accuracy of various p n expansion orders on several problems in two dimensional problems the numerical p n solutions contained negative radiation energy densities as predicted by our theorem the numerical results showed that the material temperature also became negative a result outside the scope of the theorem our numerical method can handle these negative values but they would cause problems in a radiation hydrodynamics calculation

meshless method for solving coupled radiative and conductive heat transfer in complex multi dimensional geometries a meshless method dam is employed to solve the coupled radiative and conductive heat transfer problem in a semi transparent medium enclosed in complex 2d and 3d geometries the meshless method for radiative transfer is based on the even parity formulation of the discrete ordinates method cases of combined conduction radiation are presented and the results are compared with other benchmark approximate solutions

uniformly accurate schemes for hyperbolic systems with relaxation we develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation in such systems the relaxation time may vary from order 1 to much less than unity when the relaxation time is small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results usually one cannot decouple the problem into separate regimes and handle different regimes with different methods thus it is important to have a scheme that works uniformly with respect to the relaxation time using the broadwell model of the nonlinear boltzmann equation we develop a second order scheme that works effectively with a fixed spatial and temporal discretization for all ranges of the mean free path formal uniform consistency proof for a first order scheme and numerical convergence proof for the second order scheme are also presented we also make numerical comparisons of the new scheme with some other schemes this study is motivated by the reentry problem in hypersonic computations

discretization on quadrilateral grids with improved monotonicity properties multi point flux approximation mpfa discretization methods have been applied in the oil industry since the mid 1990s the discretizations are based on a control volume formulation and the finite difference structure makes general skew grids and unstructured grids feasible in a fully implicit formulation mpfa methods are therefore suitable for flow problems in realistic reservoirs monotonicity issues are known to arise for high aspect ratios combined with skewness of computational grids for mpfa methods in this paper we improve the mpfa discretization techniques for general quadrilateral grids such that the above difficulties are handled to give a more robust discretization of the governing equations for fluid flow in porous media comparisons to the mpfa o method are made and the suggested discretization is shown to be an improvement in regards to monotonicity for smooth solutions the method performs equally well as the o method when the convergence is examined

convergence of a fully discrete scheme for two dimensional neutron transport we prove an error estimate for a fully discrete method for the numerical solution of a two dimensional model problem in neutron transport theory based on using the discrete ordinates method for the

the effects of slope limiting on asymptotic preserving numerical methods for hyperbolic conservation laws many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates an asymptotic preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit we explore the consequences of applying a slope limiter to the discontinuous galerkin dg method with linear elements for hyperbolic systems with stiff relaxation terms without a limiter the dg method gives an accurate discretization of the chapman enskog approximation of the system when the relaxation length scale is not resolved it is well known that the first order upwind or step method fails to obtain the proper asymptotic limit we show that using the minmod slope limiter also fails but that using double minmod gives the proper asymptotic limit despite its effectiveness in the asymptotic limit the double minmod limiter allows artificial extrema at cell interfaces referred to as sawteeth we present a limiter that eliminates the sawteeth but maintains the proper asymptotic limit the systems that we analyze are the hyperbolic heat equation and the p n thermal radiation equations numerical examples are used to verify our analysis

delay robustness in consensus problems we investigate the robustness of consensus schemes for linear multi agent systems mas to feedback delays to achieve this we develop a unified framework that considers linear mas models with different feedback delays e g affecting only the neighbor s output or affecting both the agent s own and its neighbors output this framework has the advantage of providing scalable simple and accurate set valued conditions for consensus using these set valued conditions previous results on consensus in mas with delays can be recovered and generalized moreover we use them to derive conditions for the convergence rate of single integrator mas with feedback delays finally building on this framework we propose a scalable delay dependent design algorithm for consensus controllers for a large class of linear mas

bounded skew high order resolution schemes for the discrete ordinates method the discrete ordinates method for the solution of the radiative heat transfer equation suffers from two main shortcomings namely ray effects and numerical smearing spatial discretization which is the cause of numerical smearing constitutes the subject of the present work bounded skew high order resolution schemes are applied to the discrete ordinate equations and compared with standard bounded high order resolution schemes clam muscl and smart as well as with the step scheme calculations are performed for two and three dimensional enclosures with transparent emitting absorbing and emitting absorbing scattering media one of the walls of the enclosure is hot while the others are cold the results demonstrate that the bounded skew high order schemes are more accurate than the bounded high order ones regardless of the radiative properties of the medium the improved accuracy is more significant for the radiation intensity along directions oblique to the coordinate lines but it is also observed for the incident radiation the difference between the results of the skewed and the standard schemes is attenuated as the optical thickness of the medium increases a drawback of the skewed schemes is their higher computational requirements associated with an increased number of iterations required for convergence

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100