We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations (LRTEs), which possess such features as multi-scale properties and high dimensionality. Utilizing the Physics-Informed Neural Networks (PINNs), we design a new adaptive exponentially weighted Asymptotic-Preserving (AP) loss function, which consists of a macroscopic auxiliary equation taking into account the diffusion limit equation explicitly. A novel property on the scheme is that the loss function gradually transfers from the transport state to the diffusion limit state as the scale parameter approaches zero. Furthermore, we combine the residual-based adaptive refinement (RAR) method with the MA-APNN method to obtain the MA-APNN-RAR method, which leads to an improvement in the accuracy and efficiency under some practical settings. A theoretical analysis about the approximation errors is given for the MA-APNN method, and numerical examples are presented to demonstrate the performance of MA-APNNs and MA-APNNs-RAR.

We propose a model-data asymptotic-preserving neural network(MD-APNN) method to solve the nonlinear gray radiative transfer equations(GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks(PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving(AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure data-driven networks in the simulation of the nonlinear non-stationary GRTEs.

The aim of this paper is to construct a new numerical scheme for the nonlinear gray radiative transfer (GRT) equations, namely, the asymptotic-preserving (AP) HNT-based unified gas kinetic scheme (UGKS). The constructed scheme is obtained by combing the UGKS for spatial discretization with the hybrid HNT method for angular discretization. Since the HNT is a hybrid angular discrete method of both PN and SN methods, the current HNT-based UGKS can not only mitigate the ray effects of the SN method largely, but also suppress the oscillations of the original PN method. Furthermore, we show that the current HNT-based UGKS also inherits the AP property of UGKS. A number of one-dimensional and two-dimensional numerical experiments are presented that validate the performance of the current scheme in both optically thin and thick regimes, as well as in mitigating the ray effects. Moreover, it can capture the initial layer solution without requiring additional treatments.

In this paper, the corrected method to the original H^T_N -unified gas kinetic scheme ( H^T_N -UGKS) is developed in order to solve the nonlinear radiative transfer equations with boundary layers. The H^T_N -UGKS is an asymptotic preserving (AP) scheme that uses UGKS for spatial discretization and the hybrid H^T_N method for angular discretization which is constructed in the paper (Li et al. in Nucl. Sci. Eng. 198(5): 993–1020, 2024). First, the correction idea in Mieussens (J. Comput. Phys. 253: 138–156, 2013) is adopted, such that H^T_N -UGKS can correctly simulate the linear radiative transfer equation with boundary layers. Then, for the nonlinear radiative transfer equations with boundary layers, the transformation from the implicit Monte Carlo (IMC) method is introduced to rewrite the nonlinear transfer equations into a linearized system. It is the key point in the construction of the current scheme to use this linearized system to construct the numerical boundary fluxes. In this way, the boundary density is included in the numerical fluxes, and consequently, the modification method for the linear radiative transfer equation can be used to deal with the nonlinear problem studied in this paper. A number of numerical examples are presented to demonstrate the accuracy and effectiveness of the current scheme for resolving boundary layers in both linear and nonlinear radiative transfer problems.

A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this paper. The discretization scheme is based on the filtered spherical harmonics (FPN) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. In order to keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo (IMC) linearization method [7] in the construction of the UGKS numerical boundary fluxes. This is an essential point in the construction. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the FPN angular discretization, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the PN reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme, called the PPFPN-based UGKS, is obtained. Furthermore, we can show that in the regime ϵ≪1 and the regime ϵ=O(1), the second-order fluxes can be simplified. And, a simplified spatial second-order scheme, called the PPFPN-based SUGKS, is thus presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preserving. By employing the FPN method for the angular variable, the proposed schemes are almost free of ray effects. Moreover, the above-mentioned way of imposing the positivity would not destroy both AP and second-order accuracy properties. To our best knowledge, this is the first time that spatial second-order, positive, asymptotic preserving and almost free of ray effects schemes are constructed for the nonlinear radiative transfer equations without operator splitting. Therefore, this paper improves our previous work on the first-order scheme [44] which could not be directly extended to high order, while keeping the solution positive. Various numerical experiments are included to validate the properties of the proposed schemes.

In our previous work, the Unified Gas Kinetic Scheme (UGKS) and an efficient time-implicit scheme (known as IUGKS) were constructed for neutron transport. Both UGKS and IUGKS have been proved to be numerical schemes with asymptotic preserve (AP) property in neutron transport simulations. However, these schemes are built based on isotropic scattering models. To extend these methods to complex engineering applications, in this paper, the IUGKS-PN scheme for anisotropic scattering models is developed firstly, and it is demonstrated that the scheme keeps good AP property in the anisotropic scattering problems. Furthermore, to relieve the complication in IUGKS-PN for high-order PN expansion, the IUGKS-AM (IUGKS with anisotropic-scattering modification) is further developed with the anisotropic scattering correction. In the IUGKS-AM, there is no need to solve the higher-order moment equation which is used in IUGKS-PN. The AP property of IUGKS-AM is also derived, while the numerical performances of IUGKS-AM is compared with IUGKS-P1 in detail. In the numerical tests, the accuracy and efficiency of IUGKS-AM are verified with typical one- and two-dimensional benchmarks, and the numerical performance of the scheme in complex problems is further verified with a three-dimensional case. The results show that the scheme can be used for scattering model with complicated functional form. This study improves the application capability of IUGKS in engineering particle transport simulations.

We propose two spatial second-order schemes for linear radiative transfer equations by using the idea of the unified gas kinetic scheme (UGKS) to construct the numerical boundary fluxes, and show that the proposed schemes are both positive and asymptotic preserving. The UGKS was proposed by Xu and Huang (J Comput Phys 229:7747–7764, 2010) for continuum and rarefied flows firstly, and was then applied to a linear radiative transfer equation by Mieussens in (J Comput Phys 253:138–156, 2013) where the asymptotic preserving property of UGKS is shown. Although it is asymptotic preserving, UGKS can not always keep the positivity of solutions. We first apply UGKS to discretize a linear radiative transfer equation to have a spatial second-order scheme. Then, by a detailed analysis of the numerical boundary fluxes, we are able to find the reasons why the positive preserving property of UGKS fails. Finally, we carefully employ a linear scaling limiter and a flux correction to make UGKS positive-preserving but still asymptotic-preserving. Consequently, we propose two spatial second-order positive and asymptotic preserving unified gas kinetic schemes for the linear radiative transfer equation, thus improving the earlier work (J Comput Phys 444:110546, 2021) where only a first-order positive and asymptotic scheme is developed. The proposed schemes can well capture the solution of the diffusion limit equation in optically thick regions without requiring the cell size being smaller than the photon’s mean free path, while the solution in optically thin regions can also be well resolved in a natural way. To our best knowledge, this is the first time that a spatial second-order positive and asymptotic preserving gas kinetic scheme for linear radiative transfer equations is constructed. Several numerical experiments are included to validate the spatial second-order accuracy, positive- and asymptotic-preserving properties of the proposed schemes.

In this paper, a high-order gas-kinetic scheme is developed for the equation of radiation hydrodynamics in equilibrium-diffusion limit which describes the interaction between matter and radiation. To recover RHE, the Bhatnagar-Gross-Krook (BGK) model with modified equilibrium state is considered. In the equilibrium-diffusion limit, the time scales of radiation diffusion and hydrodynamic part are different, and it will make the time step very small for the fully explicit scheme. An implicit-explicit (IMEX) scheme is applied, in which the hydrodynamic part is treated explicitly and the radiation diffusion is treated implicitly. For the hydrodynamics part, a time dependent gas distribution function can be constructed by the integral solution of modified BGK equation, and the time dependent numerical fluxes can be obtained by taking moments of gas distribution function. For the radiation diffusion term, the nonlinear generalized minimal residual (GMRES) method is used. To achieve the temporal accuracy, a two-stage method is developed, which is an extension of two-stage method for hyperbolic conservation law. For the spatial accuracy, the multidimensional weighted essential non-oscillation (WENO) scheme is used for the spatial reconstruction. A variety of numerical tests are provided for the performance of current scheme, including the order of accuracy and robustness.

This paper aims at the simulation of multiple scale physics for the system of radiation hydrodynamics. The system couples the fluid dynamic equations with the radiative heat transfer. The coupled system is solved by the gas-kinetic scheme (GKS) for the compressible inviscid Euler flow and the unified gas-kinetic scheme (UGKS) for the non-equilibrium radiative transfer, together with the momentum and energy exchange between these two phases. For the radiative transfer, due to the possible large variation of fluid opacity in different regions, the transport of photons through the flow system is simulated by the multiscale UGKS, which is capable of naturally capturing the transport process from the photon's free streaming to the diffusive wave propagation. Since both GKS and UGKS are finite volume methods, all unknowns are defined inside each control volume and are discretized consistently in the updates of hydrodynamic and radiative variables. For the coupled system, the scheme has the asymptotic preserving property, such as recovering the equilibrium diffusion limit for the radiation hydrodynamic system in the optically thick region, where the cell size is not limited by photon's mean free path. A few test cases, such as radiative shock wave problems, are used to validate the current approach.

In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D multi-group neutron transport equation with delayed neutron is developed. The explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order CrankNicholson scheme, become the subsets of the current IUGKS. In neutron transport, the microscopic angular flux and the macroscopic scalar flux are fully coupled in an implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step. Mathematically, the current scheme has the asymptotic preserving (AP) property in recovering automatically the diffusion solution in the continuum regime. Since the explicit scanning along neutron traveling direction within the computational domain is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other schemes, such as the explicit UGKS, the commonly used finite difference, and finite volume methods. This study shows that the IUGKS can be used faithfully to study neutron transport in practical engineering applications. AMS subject classifications: 82D75, 82C70, 82C40

This paper aims at the simulation of multiple scale physics in the system of radiation hydrodynamics. The system couples the fluid dynamic evolution equations with the radiative heat transfer. The coupled system is solved by the gas-kinetic scheme (GKS) for the compressible viscous and heat conducting flow and the unified gas-kinetic scheme (UGKS) for the non-equilibrium radiative transfer, together with the momentum and energy exchange between these two phases. For the radiative transfer, due to the possible large variation of fluid opacity in different regions, the transport of photons in the flow system is simulated by the multiscale UGKS, which is capable of naturally capturing the transport process from the free streaming to the diffusive propagation.Since both GKS and UGKS are finite volume methods,all unknowns are defined inside each control volume and are discretized consistently for the hydrodynamic and radiative variables. For the coupled system, the scheme has the asymptotical preserving (AP) property, such as recovering the equilibrium diffusion limit for the radiation hydrodynamic equations in the optically thick region, where the cell size is not limited by photon's mean free path. A few test cases, such as radiative shock wave problems, are used to validate the current approach.

In order to mitigate the ray effects of the usual discrete ordinate (S-N) method for the gray radiative transfer equations, a new numerical approach is constructed in this paper with the angular discretization by a linear finite element (FE) method and the spatial discretization by the method of unified gas kinetic scheme (UGKS). Different from the usual S-N-based UGKS for which the propagation directions of photons are discretized with finite points, the angular-FE-based UGKS considers the linear combination of all propagation directions (i.e., the unit sphere) and induces a coupling between the discrete directions. Hence, the angular-FE-based UGKS can much mitigate the ray effects to some extent as was expected, as also shown numerically by a point source problem in this paper. At the same time, it is shown that the current scheme possesses the asymptotic preserving (AP) property, that is, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photons mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. Various numerical experiments are included to validate the robustness, accuracy and AP property of the current scheme. (C) 2020 Elsevier Ltd. All rights reserved.

The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this paper, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon’s mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale-dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result.

This paper is about the construction of a unified gas-kinetic scheme (UGKS) for a coupled system of radiative transport and material heat conduction with different diffusive limits. Different from the previous approach, instead of including absorption/emission only, the current method takes both scattering and absorption/emission mechanism into account in the radiative transport process. As a result, two asymptotic limiting solutions will appear in the diffusive regime. In the strong absorption/emission case, an equilibrium diffusion limit is obtained, where the system is mainly driven by a nonlinear diffusion equation for the equilibrium radiation and material temperature. However, in the strong scattering case, a non-equilibrium limit can be obtained, where coupled nonlinear diffusion system with different radiation and material temperature is obtained. In addition to including the scattering term in the transport equation, an implicit UGKS (IUGKS) will be developed in this paper as well. In the IUGKS, the numerical flux for the radiation intensity is constructed implicitly. Therefore, the conventional CFL constraint for the time step is released. With the use of a large time step for the radiative transport, it becomes possible to couple the IUGKS with the gas dynamic equations to develop an efficient numerical method for radiative hydrodynamics. The IUGKS is a valid method for all radiative transfer regimes. A few numerical examples will be presented to validate the current implicit method for both optical thin to optical thick cases.

In order to extend the unified gas kinetic scheme (UGKS) to solve radiative transfer equations in a complex geometry, a multidimensional asymptotic preserving implicit method on unstructured mesh is constructed in this paper. With an implicit formulation, the CFL condition for the determination of the time step in UGKS can be much relaxed, and a large time step is used in simulations. Differently from previous direction-by-direction UGKS on orthogonal structured mesh, on unstructured mesh the interface flux transport takes into account multi-dimensional effect, where gradients of radiation intensity and material temperature in both normal and tangential directions of a cell interface are included in the flux evaluation. The multiple scale nature makes the UGKS be able to capture the solutions in both optically thin and thick regions seamlessly. In the optically thick region the condition of cell size being less than photon's mean free path is fully removed, and the UGKS recovers a solver for diffusion equation in such a limit on unstructured mesh. For a distorted quadrilateral mesh, the UGKS goes to a nine-point scheme for the diffusion equation, and it naturally reduces to the standard five-point scheme for a orthogonal quadrilateral mesh. Numerical computations covering a wide range of transport regimes on unstructured and distorted quadrilateral meshes will be presented to validate the current approach.

In the previous works [J. Comput. Phys. 285 (2015), 265-279 and J. Comput. Phys. 302 (2015), 222-238 ], an explicit asymptotic preserving unified gas kinetic scheme (UGKS) has been constructed for the radiative transfer equations with strong absorption/emission coefficients. In the previous UGKS, for the update of the radiation intensity in all regimes, the time step is constrained by the CFL condition, which is related to the ratio of cell size over the speed of light. In order to improve the efficiency of the scheme, an implicit UGKS (IUGKS) will be developed in this paper for numerical simulation of the unsteady radiative transport with the inclusion of scattering and absorption/emission phenomena. In IUGKS, the numerical flux for the radiation intensity is constructed implicitly, and the CFL constraint can be released. As a result, a large time step can be used in the simulation of the radiative transfer. The purpose for constructing such an implicit radiative transfer method is to couple it with the gas dynamic equations in the future for the development of numerical method for radiative hydrodynamics. Theoretically, it can be proved that the current IUGKS has the asymptotic preserving (AP) property in the strong scattering regime for the capturing of non-equilibrium diffusive limiting solutions. As a result, the IUGKS can capture accurate solutions in both the photon free transport and the diffusive limit. Actually, a smooth transition of the solution from optical thin to optical thick regions can be automatically obtained through a time-dependent evolution solution for the radiative transport across a cell interface. A few numerical examples will be presented to validate the current implicit scheme.

In this paper, we prove some blow-up criteria for the 3D Boussinesq system with zero heat conductivity and MHD system and Landau-Lifshitz equations in a bounded domain.

本文通过一个物质压力和辐射压力解耦的分子动理学BGK模型,设计了辐射流体力学方程的分子动理学数值格式.该格式有两方面的优点,一方面,由于BGK模型中流体压力和辐射压力的解耦,因而物质压与辐射压的作用可以解耦,微观方程与宏观方程间的关系变得更为简单;另一方面,利用该BGK模型,数值格式的构造与原有方法相比也大大简化.对不带扩散项的高维辐射流体力学方程组给出了我们构造的二阶BGK分子动理学格式.一维和高维的数值算例显示了新格式的性能.

The paper presents a Discontinuous Galerkin gamma-BGK (gamma-DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a gamma-BGK scheme based on WENO limiters. In this gamma-DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright (C) 2010 John Wiley & Sons, Ltd.

We present a second‐order BGK scheme for the equations of multidimensional radiation hydrodynamics (RHE) in zero diffusion limit by using the generalized Maxwellian ( Comput. Fluids 2000; 29 :917–933) and the second‐order BGK scheme for the Euler equations ( J. Comput. Phys. 1993; 109 :53–66; Gas Kinetic Scheme for Unsteady Compressible Flow Simulations . Lecture Note Series 1998‐03. Von Kárman Institute for Fluid Dynamics, 1998). This extends a (first‐order) kinetic flux vector splitting scheme (KFVS) for RHE (Tang and Wu (2000)) to a second‐order BGK scheme. Several one‐ and two‐dimensional numerical examples demonstrate improvement of the scheme in accuracy and resolution compared with the KFVS scheme (Tang and Wu (2000)) and the first‐order BGK scheme. Copyright © 2006 John Wiley & Sons, Ltd.