Standard Discrete Ordinates Method Research Articles

Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration

In this work, we describe extensions of a hybrid method for time-dependent linear, kinetic radiation transport problems to high-order time integration schemes of the diagonally-implicit Runge–Kutta (DIRK) and space–time discontinuous Galerkin (STDG) types. The hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low- and high-resolution discrete ordinates approximations are applied. The efficiency of hybrid methods constructed using DIRK, STDG, integral deferred correction (IDC), and implicit Euler schemes is compared using a test problem in one-dimensional slab geometry containing material discontinuities. It is observed that (i) higher-order methods are more efficient than the implicit Euler method, often by an order of magnitude or more; (ii) third-order methods yield solutions of a given error in roughly half the time of the second-order method of the same type; and (iii) for a given order of accuracy it is found that the most efficient class of time integration scheme is STDG, followed by DIRK, with IDC the least efficient, for the test problem considered. Two test problems in two-dimensional xy-geometry are used to compare the computational efficiency of hybrid and standard discrete ordinates methods constructed with DIRK and STDG integrators. We observe that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using angular quadratures of order 2N and N/2 for the uncollided and collided fluxes, respectively, usually reduces overall solution time by a factor of 2 or more while simultaneously reducing the resulting solution error by a factor of 2 or more for the test problems considered.

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction

In this work, we describe the implementation of an arbitrarily high-order hybrid solver for linear, kinetic, radiative transport equations. The hybrid method is derived from a splitting of the radiative flux into free-streaming and collisional components to which high- and low-resolution discrete ordinates methods are applied, respectively. Arbitrarily high orders of accuracy with respect to time and space are attained by combining an integral deferred correction (IDC) time integration scheme constructed with implicit Euler substepping on Radau II (right biased) nodes with an upwind discontinuous Galerkin (DG) spatial discretization on uniform Cartesian meshes. Numerical experiments are used to demonstrate that the aforementioned IDC methods can be constructed such that they are unconditionally stable (L-stable) to within machine precision. Asymptotic analysis is used to show that such IDC methods also preserve the diffusion limit of the underlying transport equation on the semi-discrete level, in the sense that the semi-discrete transport system under an implicit IDC discretization recovers the same IDC discretization of the limiting diffusion equation. Convergence results in one spatial dimension are provided, and it is found that while the hybrid method exhibits convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximation is comparable to a standard discrete ordinates approximation in many cases. Numerical results for two test problems in two spatial dimensions are given to compare the computational efficiency of the hybrid method against a standard discrete ordinates method, and to compare the efficiency of space–time discretizations of different orders of accuracy. The results indicate that a properly chosen hybrid discrete ordinates method can be more efficient than a standard discrete ordinates method by a factor of between 2 and 8 when IDC timestepping methods are used. Further, among the first-, third-, and fifth-order space–time discretizations considered here, the first-order method is the least efficient and the third-order method is the most efficient.

Application of alternative synthetic kernel approximation to radiative transfer in regular and irregular two-dimensional media

Alternative synthetic kernel (ASKN) approximation, just as the standard SKN method, is derived from the radiative integral transfer equations in full 3D generality. The direct and diffuse terms of thermal radiation appear explicitly in the radiative integral transfer equations as surface and volume integrals, respectively. In standard SKN method, the approximation is employed to the diffuse terms while direct terms are evaluated analytically. The alternative formulation differs from the standard one in that the direct radiation wall contributions are also approximated with the same spirit of the synthetic kernel approximation. This alternative formulation also yields a set of coupled partial differential-the ASKN–equations which could be solved using finite volume methods. This approximation is applied to radiative transfer calculations in regular and irregular two-dimensional absorbing, emitting and isotropically scattering media. Four benchmark problems-one rectangular and three irregular media-are considered, and the net radiative flux and/or incident energy solutions along the boundaries are compared with available exact, standard discrete ordinates S4 and S12, modified discrete ordinates S4, Monte Carlo and collocation spectral method to assess the accuracy of the method. The ASKN approximation yields ray effect free incident energy and radiative flux distributions, and low order ASKN solutions are generally better than those of the high order standard discrete ordinates method.

Sci‐Thur PM – Colourful Interactions: Highlights 02: A deterministic solution to the first order linear Boltzmann transport equation including magnetic fields

Purpose:To develop a novel formalism including magnetic force in the linear Boltzmann transport equation (LBTE) and to solve this equation deterministically by developing a new numerical framework.Methods:The continuity equation in six dimensions was used to derive the magnetic force term in the LBTE. The phase space variables were discretized using the multigroup method for energy variables, the discontinuous finite element method (DFEM) for the spatial variables, and using two approaches for the angular variables: the standard discrete ordinates method (DOM), and a novel angular DFEM. The calculated dose for both techniques was compared to Monte Carlo.Results:It was found that the standard source iteration approach was unstable using the DOM with magnetic fields. The Krylov solver restarted GMRES(m) overcame the instability except for cases characterized by very low media densities and large magnetic fields where convergence stagnated using smaller restart parameters. Our novel angular DFEM framework overcame these instabilities for all cases tested. Comparison with Monte Carlo showed greater than 99% of points passing a 2%/2mm gamma criterion for the DOM, and 99% of points passing for the angular DFEM.Conclusion:A novel formalism to include magnetic force within the LBTE was derived and a new numerical framework to solve the resultant equations was developed and tested using two different angular discretization techniques. Both techniques provided excellent accuracy, but the angular DFEM proved to be more stable. Dose calculations with this formalism were proven to be highly accurate, equivalent to advanced Monte Carlo algorithms.

Application of modified discrete ordinates method to combined conduction‐radiation heat transfer problems in irregular geometries

PurposeThe solution of radiative heat transfer problems in participating media is often obtained using the standard discrete ordinates method (SDOM). This method produces anomalies caused by ray effects if radiative boundary conditions have discontinuities or abrupt changes. Ray effects may be mitigated using the modified discrete ordinates method (MDOM), which is based on superposition of the solutions obtained by considering separately radiation from the walls and radiation from the medium. The purpose of this paper is to study the role of ray effects in combined conduction‐radiation problems and investigate the superiority of the MDOM over SDOM.Design/methodology/approachThe MDOM has been used to calculate radiative heat transfer in irregular geometries using body‐fitted coordinates. Here, the blocked‐off region concept, originally developed in computational fluid dynamics, is used along with the finite volume method and SDOM or MDOM to solve combined conduction‐radiation heat transport problems in irregular geometries. Enclosures with an absorbing, emitting and isotropically or anisotropically scattering medium are analyzed.FindingsThe results confirm the capability of the MDOM to minimize the anomalies due to ray effects in combined heat transfer problems, and demonstrate that MDOM is more computationally efficient than SDOM.Originality/valueThe paper demonstrates the application of MDOM to combined conduction‐radiation heat transfer problems in irregular geometries using blocked‐off method.

Analysis of the radiative transfer equation with highly assymetric phase function

This paper considers a scalar radiative transfer problem with high scattering anisotropy. Two computational methods are presented based on decomposition of the diffuse light field into a regular and anisotropic part. The first algorithm (DOMAS) singles out the anisotropic radiance in the forward scattering peak using the Small-Angle Modification of RTE. The second algorithm (DOM2+) separates the single scattering radiance as an anisotropic part, which largely defines the fine detail of the total radiance in the backscattering directions. In both cases, the anisotropic part is represented analytically. With anisotropy subtraction, the regular part of the signal, which requires a numerical solution, is essentially smoothed as a function of angles. Further, the transport equation is obtained for the regular part that contains an additional source function from the anisotropic part of the signal. This equation is solved with the discrete ordinates method. A conducted numerical analysis of this work showed that algorithm DOMAS has a strong advantage as compared to the standard discrete ordinates method for simulation of the radiance transmission, and DOM2+ is the best of the three for the reflection computations. Both algorithms offer at least a factor of three acceleration of convergence of the azimuthal series for highly anisotropic phase functions.

Application of the modified discrete ordinates method with the concept of blocked-off region to irregular geometries

The standard discrete ordinates method (SDOM) produces anomalies caused by ray effects if the temperature of the walls has discontinuities or abrupt changes. Ray effects may be mitigated using the modified discrete ordinates method (MDOM), which is based on superposition of the solutions obtained by considering separately radiation from the walls and radiation from the medium. The MDOM has been used to calculate radiative heat transfer in irregular geometries using body-fitted coordinates. Here, the blocked-off region concept is used along with the MDOM to solve radiative transfer problems in irregular geometries. Two-dimensional irregular enclosures with curved boundaries, obstacles or radiation shields and containing a participating medium are considered. The accuracy of the results calculated using the SDOM and the MDOM is assessed by comparing the predictions with those obtained by other researchers. The results confirm the capability of the MDOM to minimize the anomalies due to ray effects in various situations. It is concluded that the blocked-off region concept can be recommended as a good option to solve radiative transport problems in irregular geometries.

An accurate solver for forward and inverse transport

This paper presents a robust and accurate way to solve steady-state linear transport (radiative transfer) equations numerically. Our main objective is to address the inverse transport problem, in which the optical parameters of a domain of interest are reconstructed from measurements performed at the domain’s boundary. This inverse problem has important applications in medical and geophysical imaging, and more generally in any field involving high frequency waves or particles propagating in scattering environments. Stable solutions of the inverse transport problem require that the singularities of the measurement operator, which maps the optical parameters to the available measurements, be captured with sufficient accuracy. This in turn requires that the free propagation of particles be calculated with care, which is a difficult problem on a Cartesian grid. A standard discrete ordinates method is used for the direction of propagation of the particles. Our methodology to address spatial discretization is based on rotating the computational domain so that each direction of propagation is always aligned with one of the grid axes. Rotations are performed in the Fourier domain to achieve spectral accuracy. The numerical dispersion of the propagating particles is therefore minimal. As a result, the ballistic and single scattering components of the transport solution are calculated robustly and accurately. Physical blurring effects, such as small angular diffusion, are also incorporated into the numerical tool. Forward and inverse calculations performed in a two-dimensional setting exemplify the capabilities of the method. Although the methodology might not be the fastest way to solve transport equations, its physical accuracy provides us with a numerical tool to assess what can and cannot be reconstructed in inverse transport theory.

Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method

The Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.

A modified version of the discrete ordinates method for radiative heat transfer modelling

The standard discrete ordinates method (SDOM) suffers from two major sources of inaccuracy, the ray effects and false scattering. False scattering may be significantly reduced using high order discretization schemes, while ray effects originated from abrupt changes of wall temperatures are mitigated by the modified discrete ordinates method (MDOM). However, the MDOM is insensitive to ray effects caused by sharp gradients of the emissive power of the medium. Therefore, a new version (NMDOM) is proposed to overcome this shortcoming. The accuracy of the SDOM, MDOM and NMDOM are investigated, along with the interaction between ray effects and false scattering. It is concluded that the NMDOM yields accurate solutions even for coarse spatial and angular discretizations.

The radiative transfer solution of a rectangular enclosure using angular domain discrete wavelets

The wavelet expansion is used in order to evaluate the angular dependence of the radiative intensity in the solution of the radiative transfer equation. The radiative intensity is expanded in terms of orthogonal Daubechies’ wavelet basis in the angular domain. The method is applied to a two-dimensional rectangular enclosure with an absorbing, emitting and nonscattering medium in radiative equilibrium. One of the boundary surfaces is maintained at constant temperature T 1, while others are kept cold. This boundary conditions are chosen to demonstrate the effectiveness of the method in dealing with the geometries which are sensitive to ray effects. Centerline emissive power and surface heat flux distributions are compared well with the results given by the standard discrete ordinates method, the modified discrete ordinates method and also with the available exact solutions.