Time-dependent Transport Equation Research Articles

An Adaptive Multiresolution Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multidimensions

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multidimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is realized by error thresholding based on the hierarchical surplus, and the Runge--Kutta DG scheme is employed as the reference time evolution algorithm. We show that the scheme performs similarly to a sparse grid DG method when the solution is smooth, reducing computational cost in multidimensions. When the solution is no longer smooth, the adaptive algorithm can automatically capture fine local structures. The method is therefore very suitable for deterministic kinetic simulations. Numerical results including several benchmark tests, the Vlasov--Poisson (VP), and oscillatory VP systems are provided.

LUCKY_TD code for solving the time-dependent transport equation with the use of parallel computations

An algorithm for solving the time-dependent transport equation in the PmSn group approximation with the use of parallel computations is presented. The algorithm is implemented in the LUCKY_TD code for supercomputers employing the MPI standard for the data exchange between parallel processes.

Sparsity and level set regularization for diffuse optical tomography using a transport model in 2D

In this paper we address an inverse problem for the time-dependent linear transport equation (or radiative transfer equation) in 2D having in mind applications in diffuse optical tomography (DOT). We propose two new reconstruction algorithms which so far have not been applied to such a situation and compare their performances in certain practically relevant situations. The first of these reconstruction algorithms uses a sparsity promoting regularization scheme, whereas the second one uses a simultaneous level set reconstruction scheme for two parameters of the linear transport equation. We will also compare the results of both schemes with a third scheme which is a more traditional L2-based Landweber–Kaczmarz scheme. We focus our attention on the DOT application of imaging the human head of a neonate where the simpler diffusion approximation is not well-suited for the inversion due to the presence of a clear layer beneath the skull which is filled with ‘low-scattering’ cerebrospinal fluid. This layer, even if its location and characteristics are known a priori, poses significant difficulties for most reconstruction schemes due to its ‘wave-guiding’ property which reduces sensitivity of the data to the interior regions. A further complication arises due to the necessity to reconstruct simultaneously two different parameters of the linear transport equation, the scattering and the absorption cross-section, from the same data set. A significant ‘cross-talk’ between these two parameters is usually expected. Our numerical experiments indicate that each of the three considered reconstruction schemes do have their merits and perform differently but reasonably well when the clear layer is a priori known. We also demonstrate the behavior of the three algorithms in the particular situation where the clear layer is unknown during the reconstruction.

Cross-Section Homogenization for Reactivity-Induced Transient Calculations

ABSTRACTCurrent practice in accident simulation is to use homogenized cross-sections that are computed using a critical flux. It is shown, however, that using critically homogenized cross-sections in prompt kinetic calculations can induce large errors in the calculation (∼20% in Max Power). This work presents two alternative methods for homogenization that are shown to improve these errors (∼1% in Max Power). The first method uses a time-integrated flux from a multiphysics solution, while the second uses an asymptotic solution to the time-dependent transport equation to homogenize cross-sections. The new methods are applied to several reactivity insertions in an infinite homogeneous medium and are also shown to work well with different homogenized group structures having two, three, and six energy groups. The introduction of delayed neutron precursors is treated, including possible modifications to the asymptotic solution homogenization method. Additionally, it is observed that cross-sections computed using a critical flux may perform adequately for transients where the reactivity insertion is small (ρ < β).

Investigation of transient responses of nanoscale transistors by deterministic solution of the time-dependent BTE

In this work, the transient characteristics of nanoscale field-effect transistors (FETs) have been investigated using a deterministic solver based on the time-dependent multi-subband Boltzmann transport equation (BTE). The response to a step signal superimposed on the gate or drain electrode is simulated. The transient process can be understood as a combination of electrostatic and transport relaxation. The extracted transient relaxation time for the drain current, which is unrelated to the direct-current (DC) shift, is important for transient device modeling.

Modulated heat pulse propagation and partial transport barriers in chaotic magnetic fields

Direct numerical simulations of the time dependent parallel heat transport equation modeling heat pulses driven by power modulation in 3-dimensional chaotic magnetic fields are presented. The numerical method is based on the Fourier formulation of a Lagrangian-Green's function method that provides an accurate and efficient technique for the solution of the parallel heat transport equation in the presence of harmonic power modulation. The numerical results presented provide conclusive evidence that even in the absence of magnetic flux surfaces, chaotic magnetic field configurations with intermediate levels of stochasticity exhibit transport barriers to modulated heat pulse propagation. In particular, high-order islands and remnants of destroyed flux surfaces (Cantori) act as partial barriers that slow down or even stop the propagation of heat waves at places where the magnetic field connection length exhibits a strong gradient. Results on modulated heat pulse propagation in fully stochastic fields and across magnetic islands are also presented. In qualitative agreement with recent experiments in LHD and DIII-D, it is shown that the elliptic (O) and hyperbolic (X) points of magnetic islands have a direct impact on the spatio-temporal dependence of the amplitude of modulated heat pulses.

Numerical Modeling of Photon Migration in Human Neck Based on the Radiative Transport Equation

Biomedical optical imaging has a possibility of a comprehensive diagnosis of thyroid cancer in conjunction with ultrasound imaging. For improvement of the optical imaging, this study develops a higher order scheme for solving the time-dependent radiative transport equation (RTE) by use of the finite-difference and discrete-ordinate methods. The accuracy and efficiency of the developed scheme are examined by comparison with the analytical solutions of the RTE in homogeneous media. Then, the developed scheme is applied to describing photon migration in the human neck model. The numerical simulations show complex behaviors of photon migration in the human neck model due to multiple diffusive reflection near the trachea.

Ignition curves for deuterium/helium-3 fuel in spherical tokamak reactor

In this paper, ignition curve for deuterium /helium-3 fusion reaction is studied. Four fusion reactions are considered. Zero-dimensional model for the power balance equation has been used. The closed ignition curves for ρ=constant (ratio of particle to energy confinement time) have been derived. The results of our calculations show that ignited equilibria for deuterium /helium-3 fuel in a spherical tokamak is only possible for ρ= 5.5 and 6. Then, by using the energy confinement scaling and parameters of the spherical tokamak reactor, the plasma stability limits have been obtained in n e, T plane and, to determine the thermal instability of plasma, the time-dependent transport equations have been solved.

Dynamic Monte Carlo calculation method by solving frequency domain transport equation using the complex-valued weight Monte Carlo method

A Monte Carlo method that calculates the transient behavior of neutron flux in the frequency domain is proposed in this study. A time-dependent neutron transport equation, in which the neutron source term varies over time while other properties remain constant, is Fourier transformed to obtain the transport equation in the frequency domain. The complex-valued transport equation in the frequency domain is subsequently solved with the complex-valued weight Monte Carlo method for each frequency contained in the Fourier transformed source intensity. The effect of delayed neutrons can be easily included in this frequency domain transport equation. Using the inverse Fourier transformation of the neutron flux in the frequency domain, we can obtain the time variation of the flux. This method is applicable to transient analyses of a subcritical system with a time-varying neutron source intensity. Several numerical examples indicate that the newly developed frequency domain calculation method provides good results compared to the time-dependent calculation method in the time domain. The computation time in the frequency domain is significantly shorter than that in the time domain, particularly for a nearly critical system.

Nonlinear dynamical behavior of Xenon atoms along dislocation lines in UO2+x nuclear fuel

Experimental results showed that there are a few Xenon atom bubbles connected by the dislocation line in the UO2+x nuclear fuel, and the largest radius of bubbles is about 45 nm. This phenomenon is in contrast to traditional bubble formation mechanism. This phenomenon is very important in understanding the properties of nuclear fuel. In this work, we apply a time-dependent microscopic atom transport equation and take into account stress coherent potential in the boundary of the dislocation. Using the equation, we numerically solved the stress coherence effect and studied the transfer properties of Xenon atoms along the dislocation line. Our numerical results show that the transport of the Xenon atoms along the dislocation changes nonlinearly with the external driving energy, and reaches at the saturation values. It explains the growth limit of Xenon atom bubbles that is in agreement with the experiment results.

Two-dimensional equations of the surface harmonics method for solving problems of spatial neutron kinetics in square-lattice reactors

Two-dimensional time-dependent finite-difference equations of the surface harmonics method (SHM) for the description of the neutron transport are derived for square-lattice reactors. These equations are implemented in the SUHAM-TD code. Verification of the derived equations and the developed code are performed by the example of known test problems, and the potential and efficiency of the SHM as applied to the solution of the time-dependent neutron transport equation in the diffusion approximation in two-dimensional geometry are demonstrated. These results show the substantial advantage of SHM over direct finite-difference modeling in computational costs.

"Shell" approach to modeling of impurity spreading from localized sources in plasma

In fusion devices, strongly localized intensive sources of impurities may arise unexpectedly or can be created deliberately through impurity injection. The spreading of impurities from such sources is essentially three-dimensional and nonstationary phenomenon involving physical processes of extremely different time scales. Numerical modeling of such events is still a very challenging task even by using most modern computers. To diminish the calculation time drastically a "shell" model has been elaborated that allows to reduce equations for particle, parallel momentum and energy balances of various ion species to one-dimensional equations describing the time evolution of radial profiles for several most characteristic parameters. The assumptions of the "shell" approach are verified by comparing its predictions with a numerical solution of one-dimensional time-dependent transport equations.

An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation

We propose a semi-Lagrangian numerical algorithm for a time-dependent, anisotropic temperature transport equation in magnetized plasmas in regimes with negligible variation of the magnitude of the magnetic field B along field lines. The approach is based on a formal integral solution of the parallel (i.e., along the magnetic field) transport equation with sources. While this study focuses on a Braginskii (local) heat flux closure, the approach is able to accommodate nonlocal parallel heat flux closures as well. The numerical implementation is based on an operator-split formulation, with two straightforward steps: a perpendicular transport step (including sources), and a Lagrangian (field-line integral) parallel transport step. Algorithmically, the first step is amenable to the use of modern iterative methods, while the second step has a fixed cost per degree of freedom (and is therefore algorithmically scalable). Accuracy-wise, the approach is free from the numerical pollution introduced by the discrete parallel transport term when the perpendicular to parallel transport coefficient ratio χ⊥/χ∥ becomes arbitrarily small, and is shown to capture the correct limiting solution when ϵ=χ⊥L∥2χ∥L⊥2→0 (with L∥, L⊥ the parallel and perpendicular diffusion length scales, respectively). Therefore, the approach is asymptotic-preserving. We demonstrate the performance of the scheme with several numerical experiments with varying magnetic field complexity in two dimensions, including the case of heat transport across a magnetic island in cylindrical geometry in the presence of a large guide field.

A Multilevel Projective Method for Solving the Space-Time Multigroup Neutron Kinetics Equations Coupled with the Heat Transfer Equation

We present a computational method for adequate and efficient coupling of the multigroup neutron transport equation with the precursor and heat transfer equations. It is based on the multilevel nonlinear quasi-diffusion (QD) method for solving the multigroup transport equation. The system of equations includes the time-dependent high-order transport equation and time-dependent multigroup and effective one-group low-order QD equations. We also apply the α-approximation for the time-dependent high-order transport equation. This approach enables one to avoid storing the angular flux from the previous time step. Numerical results for model transient problems are presented.

A Coarse-Mesh Method for the Time-Dependent Transport Equation

A new solution technique is derived for the time-dependent transport equation. This approach extends the steady-state coarse-mesh transport method that is based on global-local decompositions of large (i.e., full-core) neutron transport problems. The new method is based on polynomial expansions of the space, angle, and time variables in a response-based formulation of the transport equation. The local problem (coarse-mesh) solutions, which are entirely decoupled from each other, are characterized by space-, angle-, and time-dependent response functions. These response functions are, in turn, used to couple an arbitrary sequence of local problems to form the solution of a much larger global problem. In the current work, the local problem (response function) computations are performed using the Monte Carlo method, while the global (coupling) problem is solved deterministically. The spatial coupling is performed by orthogonal polynomial expansions of the partial currents on the local problem surfaces, and similarly, the time-dependent response of the system (i.e., the time-varying flux) is computed by convolving the time-dependent surface partial currents and time-dependent volumetric sources against precomputed time-dependent response kernels.

Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Time-dependent equations of the surface harmonics method (SHM) are obtained for planar one-dimensional geometry. The equations are verified by calculations of test problems from Benchmark Problem Book ANL-7416, and the capabilities and efficiency of applying the SHM for solving the time-dependent neutron transport equation in the diffusion approximation are demonstrated. The results of the work show that the implementation of the SHG for full-scale computations will make possible substantial progress in the efficient solution of time-dependent problems of neutron transport in nuclear reactors.

An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties

In the present work the absorption and reduced scattering coefficients of the homogeneous scattering medium are determined using three various types of the diffusion model (classical, refined and extrapolated) in the case of the scattering slab geometry. The expression for the refined diffusion model in this case was derived for the first time. These coefficients are determined from experimental temporal distributions of transmitted photons at different slab thickness (from 10mm to 110mm) with fitting procedure. The specific dependencies of the scattering and absorption coefficients on the slab thickness were obtained for each of the examined models, although it is physically impossible for the same substance. This effect is caused by the fact that the significant features of the scattering indicatrix are lost when the diffusion approximation has been applied to the time-dependent radiation transport equation. This explanation was confirmed by Monte Carlo simulation, where the similar dependencies were obtained. There are presented several remarks regarding the experimental measurements when diffusion models are used.

Numerical solution of the time dependent neutron transport equation by the method of the characteristics

This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.

Green's function of the time-dependent radiative transport equation in terms of rotated spherical harmonics

The time-dependent radiative transport equation is solved for the three-dimensional spatially uniform infinite medium which is illuminated by a point unidirectional source using a spherical harmonics transform under rotation. Apart from the numerical evaluation of a spherical Hankel transform which connects the spatial distance with the radial distance in Fourier space, the dependence on all variables is found analytically. For the special case of a harmonically modulated source, even the spherical Hankel transform can be carried out analytically. Additionally, a special solution for the isotropically scattering infinite medium is given. The Monte Carlo method is used for a successful verification of the derived solution.

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry: Part II: Transient simulation

In an earlier paper we utilized a novel method using cellular neural networks (CNNs) coupled with spherical harmonics method to solve the steady state neutron transport equation in x–y geometry. Here, the previous work is extended to the study of time-dependent neutron transport equation. To achieve this goal, an equivalent electrical circuit based on a second-order form of time-dependent neutron transport equation and one equivalent group of neutron precursor density is obtained by the CNN method. The CNN model is used to simulate step and ramp perturbation transients in a typical 2D core.