Time-dependent Transport Equation Research Articles

Development of multi-physics calculation method in lead cooled fast reactor code system MOSASAUR

A full-core three-dimensional multi-physics calculation method was researched based on the Lead cooled fast reactor (LFR) code system MOSASAUR, which is focused on both core steady-state and transient analyses of LFR. In the previous version of MOSASAUR, cross-sections generation module and core steady-state simulation module have been developed. In this research, the stiffness confinement method is employed to solve the time-dependent multi-group neutron transport equation to expand the capabilities of core transient analyses. And the thermal hydraulic calculation method is developed as thermal feedback module for core steady-state and transient analyses. Based on the fixed-point iteration scheme, the neutron module and thermal hydraulic module are coupled at each time step. The LMW benchmark and the problem based on ORAL 19-rod bundle are employed to verify the accuracy of transient calculation and thermal hydraulic module respectively. Finally, the multi-physics calculation method is utilized to simulate the transient process of the MicroURANUS core. The simulation results demonstrate the capability of the constructed multi-physics calculation framework to accurately simulate the transient behaviors of LMRs. Numerical results showed the good accuracy of the newly-developed multi-physics calculation module with MOSASAUR.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Physics-informed neural networks for solving time-dependent mode-resolved phonon Boltzmann transport equation

The phonon Boltzmann transport equation (BTE) is a powerful tool for modeling and understanding micro-/nanoscale thermal transport in solids, where Fourier’s law can fail due to non-diffusive effect when the characteristic length/time is comparable to the phonon mean free path/relaxation time. However, numerically solving phonon BTE can be computationally costly due to its high dimensionality, especially when considering mode-resolved phonon properties and time dependency. In this work, we demonstrate the effectiveness of physics-informed neural networks (PINNs) in solving time-dependent mode-resolved phonon BTE. The PINNs are trained by minimizing the residual of the governing equations, and boundary/initial conditions to predict phonon energy distributions, without the need for any labeled training data. The results obtained using the PINN framework demonstrate excellent agreement with analytical and numerical solutions. Moreover, after offline training, the PINNs can be utilized for online evaluation of transient heat conduction, providing instantaneous results, such as temperature distribution. It is worth noting that the training can be carried out in a parametric setting, allowing the trained model to predict phonon transport in arbitrary values in the parameter space, such as the characteristic length. This efficient and accurate method makes it a promising tool for practical applications such as the thermal management design of microelectronics.

Capturing the diffusive behavior of the multiscale linear transport equations by Asymptotic-Preserving Convolutional DeepONets

In this paper, we introduce two types of novel Asymptotic-Preserving Convolutional Deep Operator Networks (APCONs) designed to solve the multiscale time-dependent linear transport equations. We observe that the vanilla physics-informed DeepONets with modified MLP may exhibit instability in maintaining the desired limiting macroscopic behavior. Therefore, this necessitates the utilization of an asymptotic-preserving loss function. Drawing inspiration from the heat kernel in the diffusion equation, we propose a new architecture called Convolutional Deep Operator Networks, which employs multiple local convolution operations instead of a global heat kernel, along with pooling and activation operations in each filter layer. Our APCON methods possess a parameter count that is independent of the grid size and are capable of capturing the diffusive behavior of the linear transport problem. Finally, we validate the effectiveness of our methods through several numerical examples.

Electromagnetic Cascade Emission from Neutrino-coincident Tidal Disruption Events

The potential association between Tidal Disruption Events and high-energy astrophysical neutrinos implies the acceleration of cosmic rays. These accelerated particles will initiate electromagnetic (EM) cascades spanning from keV to GeV energies via the processes related to neutrino production. We model the EM cascade and neutrino emissions by numerically solving the time-dependent transport equations, and discuss the implications for AT2019dsg and AT2019fdr in the X-ray and γ-ray bands. We show that the γ-ray constraints from Fermi can constrain the size of the radiation zone and the maximum energy of injected protons, and that the corresponding expected neutrino event numbers in follow-up searches are limited to be less than about 0.1. Depending on the efficiency of p γ interactions and the time at which the target photons peak, the X-ray and γ-ray signals can be expected closer to the peak of the optical-ultraviolet luminosity or to the time of the neutrino production.

High-Fidelity Transient Analysis with STREAM Based on the Predictor-Corrector Quasi-Static Method

The predictor-corrector quasi-static method (PCQM) is used to solve the transient problem in the STREAM code, a steady-state and transient reactor analysis code with the method of characteristics. In PCQM, the angular neutron flux undergoes a factorized split to form the product of shape and amplitude functions. The time-dependent neutron transport equation is solved to obtain the shape function whereas the amplitude function is obtained by resolving the exact point kinetics equations (EPKEs). A two-level coarse mesh finite difference technique is implemented to reduce the transient running time of the transport solution. Moreover, high-order polynomial interpolation is applied to the kinetics parameters utilized in EPKEs to reduce the error when the reactivity insertion is nonlinear. Several numerical benchmarks are solved to justify the application of the procedure, proving that the method maintains solution accuracy.

A Data-driven, Physics-based Transport Model of Solar Energetic Particles Accelerated by Coronal Mass Ejection Shocks Propagating through the Solar Coronal and Heliospheric Magnetic Fields

In an effort to develop computational tools for predicting radiation hazards from solar energetic particles (SEPs), we have created a data-driven physics-based particle transport model to calculate the injection, acceleration, and propagation of SEPs from coronal mass ejection (CME) shocks traversing through the solar corona and interplanetary magnetic fields. The model runs on an input of corona and heliospheric plasma and magnetic field configuration from a magnetohydrodynamic model driven by solar photospheric magnetic field measurements superposed with observed CME shocks determined from coronagraph images. SEP source particles are injected at the shock using the result of diffusive shock acceleration formulation from a characteristic obliquity-dependent injection from a heated solar wind thermal tail population. With several advanced computation techniques involving stochastic simulation and integration, the model obtains the particle intensity at any location in interplanetary space through the rigorous solution to the time-dependent 5D focus transport equation in the phase space that includes perpendicular diffusion. We apply the model to the 2011 November 3 CME event. The calculation results reproduce multispacecraft SEP observations at Earth and STEREO-B reasonably well without normalization of particle flux. The observations at STEREO-A can be reproduced by rescaling particle energy or modified energy dependence of particle diffusion coefficients. This circumsolar SEP event seen by spacecraft at Earth, STEREO-A, and STEREO-B at widely separated longitudes can be explained by diffusive shock acceleration by a single CME shock with a moderate speed.

Continuous-Energy Time-Dependent Coarse Mesh Transport (COMET) Method for Kinetics Calculations

In this paper, the novel continuous-energy coarse mesh transport (COMET) method is extended to perform time-dependent neutronics calculations in highly heterogeneous reactor core problems. In this method, the time-dependent transport equation is converted into a series of steady-state transport equations by estimating the time derivative term using implicit finite differencing. The resulting steady-state transport equations, having additional terms that are imbedded in the total collision term and in the volumetric source terms, are then solved by the steady-state COMET method, in which all the phase-space variables, including energy, are treated continuously. Finally, the fission density distribution constructed by the steady-state COMET is used to solve a set of ordinary differential equations to obtain the delayed neutron precursor concentrations. The time-dependent COMET method is benchmarked against a direct continuous-energy Monte Carlo method (i.e., MCNP) in a set of infinite homogeneous problems and a set of single-assembly benchmark problems consisting of identical pin cells. It is found that the COMET results agree very well with the Monte Carlo reference solutions while maintaining its formidable computational speed (orders of magnitude faster than the Monte Carlo method).

Machine learning assisted two-phase upscaling for large-scale oil-water system

The computation of two-phase upscaled functions entails solving time-dependent flow and transport equations over target regions, which is usually the most time-demanding component in the overall two-phase upscaling procedure. For large-scale reservoir models with a great number of coarse grid blocks, it can be very computationally expensive to calculate the two-phase upscaled functions for each individual coarse block. To address this problem, we develop a machine learning assisted upscaling (MLAU) approach, in which the two-phase upscaling is only performed for representative coarse blocks selected by a convolutional neural network (CNN) based clustering model, while the two-phase upscaled functions are quickly predicted for the rest of the coarse blocks using a regression algorithm. The performance of MLAU approach was assessed with three cases involving Gaussian, channelized and SPE 10 sector models, respectively. Numerical results have shown that the MLAU approach consistently provides coarse-scale results with close agreement with the results using full flow-based upscaling. Because two-phase numerical upscaling is only applied for representative coarse blocks (about 5% in each case), the speedups relative to the full flow-based upscaling are significant, ranging from 6.2 to 13.5. Compared to the fine-scale simulations, the speedups range from 27.0 to 47.2.

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.

Asymptotic-Preserving Neural Networks for Multiscale Time-Dependent Linear Transport Equations

In this paper we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINN) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are directly fit for the neural network approach. We construct a micro-macro decomposition based neural network, and also build in a mass conservation mechanism into the loss function, in order to capture the dynamic and multiscale nature of the solutions. Numerical examples are used to demonstrate the effectiveness of this APNNs.

Theoretical Study for Carrier Transit Limited Performance of Gate-All-Around Si Nanowire Transistor by Time-Dependent Quantum Transport Simulation

Theoretically, the upper limit performance of nanoscale transistor should be limited by carrier transit time through the channel as the charge in channel cannot follow the gate voltage and the gate capacitance will show strong frequency dependence if the gate voltage varies faster than the carrier transit time. Currently, the operation frequency of nanoscale transistor is limited by the non-ideal factors such as parasitic effects, and it is far below the limit caused by carrier transit time. However, how good the transistor can be if one can minimize these non-ideal factors is still a problem, thus, it would be worth exploring its upper limit performance. In this article, time-dependent quantum transport simulation is performed to explore the carrier transit limited performance of the gate-all-around (GAA) Si nanowire transistor. By using the mode space approach, the 3-D time-dependent quantum transport equation is decoupled to a 2-D confinement problem and a 1-D transport problem, and then self-consistently solved with the Poisson’s equation. The transient characteristics of channel charge are investigated, and it shows the response time of channel charge is in the scale of 0.1 ps. The responses of channel charge and drain current to high-frequency gate voltages are simulated. The carrier transit limited performance including the frequency-dependent gate capacitance and transconductance, 3 dB bandwidth, and cutoff frequency, are calculated and discussed.

A sweep-based low-rank method for the discrete ordinate transport equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via the “unconventional” basis update & Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

Application of stiffness confinement method within variational nodal method for solving time-dependent neutron transport equation

The stiffness confinement method (SCM) is implemented in the variational nodal method (VNM) framework to solve time-dependent multi-group neutron transport equations. With frequency-transform, the time-dependent neutron equation is decoupled from the precursor equation and transformed into a dynamic eigenvalue problem which can be solved with the in-house VNM code VITAS. An improved time-source iteration method within the VNM-SCM framework is proposed to accelerate constructing response matrices and reduces memory usage. Based on the VNM-SCM scheme, the kinetics module of VITAS is developed for the Cartesian and hexagonal-z geometry and verified by several benchmark problems.Numerical results demonstrate that VNM-SCM can perform accurate and efficient transient calculations with large time-step sizes. For the two-dimensional TWIGL benchmark, the VNM-SCM produces a maximum core power error of -0.57% with the time-step size of 20 ms. For the three-dimensional LMW benchmark, the VNM-SCM results in a maximum core power error of 0.95% with the time-step size of 2 s. The VNM-SCM also yields an accurate neutron flux prediction for the hexagonal-z geometry problem. In addition, the effect of the angular expansion is investigated systematically. Examinations with the three-dimensional LMW benchmark display an error reduction in the core power error from 0.131% to 0.011% from P1 to P3 interfacial angular expansion, with a substantial computing time increase of 6.14 hours. The study indicates that the high-order angular expansion significantly increases the computing time but contributes little to the accuracy of the core power. Besides, it was found that the initial steady-state calculation dominates the shape error of the flux in the transient calculation.

An efficient collocation method for long-time simulation of heat and mass transport on evolving surfaces

This paper presents an efficient collocation method which combines the generalized finite difference method (GFDM) with the Krylov deferred correction (KDC) method for the long-time simulation of heat and mass transport on evolving surfaces. The KDC method utilizes a pseudo-spectral-type temporal collocation formulation to discretize the time-dependent surface heat and mass transport equation in each time marching step, where the time derivatives at the collocation points are introduced as the new unknown variables. A low-order time marching scheme is then applied as an effective preconditioner in the Jacobian-Free Newton-Krylov framework to decouple the spatial surface PDEs at different collocation nodes. Each decoupled surface PDE is then solved by the meshless GFDM, where both the continuous-form evolving surfaces defined by parametric equations and discretized-form evolving surfaces composed of point clouds are considered in the GFDM spatial discretization. Numerical experiments show that the combined GFDM-KDC solver is a promising numerical scheme for long-time evolution simulation of heat and mass transport on intractable evolving surfaces.

Stochastic radiative transfer in random media. II. Coupling of radiation to material.

We study the mechanism of the impact of random media on the stochastic radiation transport based on a one-dimensional (1D) planar model. To this end, we use a random sampling of mixtures combined with a deterministic solution of the time-dependent radiation transport equationcoupled to a material temperature equation. Compared to purely absorbing cases [C.-Z. Gao etal., Phys. Rev. E 102, 022111 (2020)10.1103/PhysRevE.102.022111], we find that material temperatures can significantly suppress the impact of mixing distribution and size, which is understood from the analysis of energy transport channels. By developing a steady-state stochastic transport model, it is found that the mechanism of transmission of radiation is distance dependent, which is closely related to the mean free path of photons l_{p}. Furthermore, we suggest that it is the relationship between l_{p} and L (the width of random medium) that determines the impact of random media on the stochastic radiation transport, which is further corroborated by additional simulations. Most importantly, combining the proposed simple relationship and 1D simulations, we resolve the existing disputable issue of the impact of random media in previous multidimensional works, showing that multidimensional results are essentially consistent and the observed weak or remarkable impact of random media is mainly due to the distinctly different relationship between l_{p} and L. Our results may have practical implications in relevant experiments of stochastic radiative transfer.

A Sweep-Based Low-Rank Method for the Discrete Ordinate Transport Equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via an unconventional basis update and Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

Analytical solution of a non-homogeneous boundary-value problem for the transport equation in an earth to air energy exchanger by initial base analysis method

In this paper, a one-dimensional time-dependent transport equation with inhomogeneous Dirichlet boundary conditions, often encountered in the study of an Earth to Air Energy Exchanger, is solved by a practical analytical method. The initial condition function is not randomly set to verify the transport equation at the initial time, but determined as a first approximation to the solution of the transport equation (at t=0). As in the homotopy analysis method, the solution of the transport equation is expressed by the set of a basis function constructed with the obtained initial condition function. The components of the solution in this base, are time’s functions deduced from the boundary conditions of the problem. Suitable endings are noticeable between the proposed model and experimental measurements.

A Stable Method for Solving the Simplified Time-Dependent Neutron Transport Equation with Background Motion

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio λ is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

An Implicit Finite Volume Scheme to Solve the Time-dependent Radiation Transport Equation Based on Discrete Ordinates

Abstract We describe a new algorithm to implicitly solve the time-dependent, frequency-integrated radiation transport (RT) equation, which is coupled to an explicit solver for equations of magnetohydrodynamics (MHD) using Athena++. The radiation field is represented by specific intensities along discrete rays, which are evolved using a conservative finite volume approach for both Cartesian and curvilinear coordinate systems. All terms for spatial transport of photons and interactions between gas and radiation are calculated implicitly together. An efficient Jacobi-like iteration scheme is used to solve the implicit equations. This removes any time-step constraint due to the speed of light in RT. We evolve the specific intensities in the lab frame to simplify the transport step. The lab frame specific intensities are transformed to the comoving frame via Lorentz transformation when the source term is calculated. Therefore, the scheme does not need any expansion in terms of v/c. The radiation energy and momentum source terms for the gas are calculated via direct quadrature in the angular space. The time step for the whole scheme is determined by the normal Courant–Friedrichs–Lewy condition in the MHD module. We provide a variety of test problems for this algorithm, including both optically thick and thin regimes, and for both gas and radiation pressure-dominated flows to demonstrate its accuracy and efficiency.