Radiation Diffusion Equations Research Articles

A Selectively Relaxed Splitting Preconditioning Strategy for the Flux-Limited Multi-Group Radiation Diffusion Equations in Three Dimensions

A Selectively Relaxed Splitting Preconditioning Strategy for the Flux-Limited Multi-Group Radiation Diffusion Equations in Three Dimensions

High-precision meshfree methods for solving 3D radiation diffusion problem in irregular spaces

Radiation diffusion is a significant phenomenon in inertial confinement fusion, geology and so on. Solving nonlinear radiation diffusion equation in irregular spaces is challenging. In this paper, the direct linearisation algorithm and the successive permutation iteration algorithm are constructed for solving 3D radiation diffusion equation, which are full-implicit discretization on time. The high-precision meshfree methods are provided by the Kansa's non-symmetric collocation method with the compactly supported radial basis function. Ultimately, the accuracy and efficiency of the presented algorithms are verified through the comparison of the fixed point, Newton and SOR methods in 3D numerical experiments. These novel meshfree methods not only avoid the complexity of grid generation, but also provide high-precision numerical solution for nonlinear radiation diffusion equation.

A Generalized Selectively Relaxed Matrix Splitting Preconditioning Strategy for Three-Dimensional Flux-Limited Multi-Group Radiation Diffusion Equations

A Generalized Selectively Relaxed Matrix Splitting Preconditioning Strategy for Three-Dimensional Flux-Limited Multi-Group Radiation Diffusion Equations

Meshfree methods for nonlinear equilibrium radiation diffusion equation with jump coefficient

Meshfree methods for nonlinear equilibrium radiation diffusion equation with jump coefficient

Three-Level Scheme for Solving the Radiation Diffusion Equation

Three-Level Scheme for Solving the Radiation Diffusion Equation

A fully asymptotic preserving decomposed multi-group method for the frequency-dependent radiative transfer equations

The opacity of FRTE depends on not only the material temperature but also the frequency, whose values may vary several orders of magnitude for different frequencies. The gray radiation diffusion and frequency-dependent diffusion equations are two simplified models that can approximate the solution to FRTE in the thick opacity regime. The frequency discretization for the two limit models highly affects the numerical accuracy. However, classical frequency discretization for FRTE considers only the absorbing coefficient. In this paper, we propose a new decomposed multi-group method for frequency discretization that is not only AP in both gray radiation diffusion and frequency-dependent diffusion limits, but also the frequency discretization of the limiting models can be tuned. Based on the decomposed multi-group method, a full AP scheme in frequency, time, and space is proposed. Several numerical examples are used to verify the performance of the proposed scheme.

A class of new successive permutation iterative algorithms with the relaxation factor for radiation diffusion equation

A class of new successive permutation iterative algorithms with the relaxation factor for radiation diffusion equation

An Algebraic Multigrid-Based Physical Factorization Preconditioner for the Multi-Group Radiation Diffusion Equations in Three Dimensions

An Algebraic Multigrid-Based Physical Factorization Preconditioner for the Multi-Group Radiation Diffusion Equations in Three Dimensions

Meshfree methods for two-dimensional three-temperature radiation diffusion equations

Two-dimensional three-temperature radiation diffusion equations coupled with the temperatures of photon, electron and ion are widely used in the numerical research of laser-driven implosion in inertial confinement fusion experiments. The equations describe the radiation evolution of electron, ion and photon among various materials. So they have strong nonlinearity and are built on complicated computational domains. Traditional numerical methods such as finite difference method and finite element method rely on meshes for discretization. In this paper, two types of meshfree methods based on radial basis function are proposed to solve two-dimensional three-temperature radiation diffusion equations. The first is using direct linearization of the diffusion term on each time layer and using non-symmetric collocation method combined with the compactly supported radial basis function on space. The second is using non-symmetric collocation method on space but constructing a successive permutation iteration algorithm with fully implicit scheme on each time layer. By comparing different linearization methods, we can find that successive permutation iteration algorithm is more effective. Accurate performance of the proposed methods are confirmed by a large number of numerical experiments.

Meshfree methods for nonlinear equilibrium radiation diffusion equation

The radiation diffusion problem is usually applied to astrophysics and inertial confinement fusion. It seems that the traditional numerical methods such as finite difference method and finite element method is not efficiency to solve the problem because of its nonlinearity and complex computational domain. In this paper, some fast meshfree methods based on radial basis function are proposed to solve the equilibrium radiation diffusion equation. Because the modeling problem has two nonlinear terms, so we use the successive permutation iteration and linearization methods to approximate the nonlinear terms. Firstly, diffusion terms of the one-dimensional (1D) and two-dimensional (2D) equilibrium radiation diffusion equations are linearized by constructing a successive permutation iteration method for fully-implicit discretization. Then, three kinds of different linearization methods (factorization iteration, Picard–Newton iteration and Richtmyer iteration) are proposed and used to linearization of other nonlinear terms. Finally, the linearized equation is solved numerically by Kansa’s method with compactly supported radial basis function. The reliability, accuracy and effectiveness of new methods are verified by 1D and 2D numerical experiments. The advantages of presented method are apparent. They require neither domain nor boundary discretization, and no domain integration is required. New methods are easy for implementation and coding. They not only avoid the complexity of generation mesh, but also solve the equilibrium radiation diffusion equation with high accuracy.

Fast meshfree methods for nonlinear radiation diffusion equation

Radiation diffusion is a phenomenon of interest in the field of astrophysics, inertial confinement fusion and so on. Since it is modeled by nonlinear equations that are usually solved in complex domains, it is difficult to solve by means of finite element method and finite difference method and so on. In the paper, we will provide a kind of new fast meshfree methods based on radial basis functions. At first, the part of diffusion terms for 1D and 2D radiation diffusion equations are linearized directly on time to form the new implicit schemes, and Kansa’s non-symmetric collocation method with the compactly supported radial basis function is used to solve the radiation diffusion problem. Second, the successive permutation iterative algorithms for full-implicit discretization on time are constructed furtherly, which are more efficient than the former algorithm. In the end, the accuracy and efficiency of the presented algorithms are verified by 1D and 2D numerical experiments. The new meshfree methods not only avoid the complexity of mesh generation, but also solve the radiation diffusion problem with high accuracy.

Supersonic–subsonic transition region in radiative heat flow via self-similar solutions

We study the radiative hydrodynamics flow of radiation-driven heat waves in hot dense plasmas, using approximate self-similar solutions. Specifically, we have focused on the intermediate regime between the pure radiative supersonic flow and the pure subsonic regime. These two regimes were investigated using both exact self-similar solutions and numerical simulations; however, most of the study used numerical simulations, mainly because the radiative heat wave and the shock regions are not self-similar altogether. In a milestone work [Garnier et al., “Self-similar solutions for a nonlinear radiation diffusion equation,” Phys. Plasmas 13, 092703 (2006)], it was found that for a specific power law dependency temperature profile, a unique exact self-similar solution exists that is valid for all physical regimes. In this work, we approximate Garnier's exact solution for a general power-law temperature-dependency, using simple analytical considerations. This approximate solution yields a good agreement compared to numerical simulations for the different thermodynamic profiles within the expected range of validity. In addition, we offer an approximate solution for the energies absorbed in the matter, again, for a general power-law temperature profile. Our approximate self-similar solution for the energy yields very good results compared to exact numerical simulations for both gold and Ta2O5. We also set a comparison of our self-similar solutions with the results of an experiment for radiation temperature measurement in a Hohlraum in low-density foams that is addressed directly to the intermediate regime, yielding a good agreement and similar trends. The different models as well as the numerical simulations are powerful tools to analyze the supersonic–subsonic transition region.

The VEM-based positivity-preserving conservative scheme for radiation diffusion problems on generalized polyhedral meshes

In this paper, for the 3D radiation diffusion equation on generalized polyhedral meshes, we construct a positivity-preserving conservative scheme based on the virtual element method (VEM). Firstly, the equation is discretized by finite volume, and the one-sided flux on the cell-face is discretized by using the fixed stencil of all vertices. Then, the nonlinear two-point flux approximation is used to obtain the equations about cell-centered unknowns and vertex unknowns; the lowest order VEM solves the vertex values of the mesh. Finally, the conservative equations containing only cell-centered unknowns are solved. This scheme ensures the high accuracy of VEM and embodies the local conservation and positivity-preserving. Compared with the existing nonlinear positivity-preserving finite volume schemes, the new scheme does not need nonlinear iteration for linear problems. For nonlinear problems, it only needs to use the VEM to solve the nonlinear system. Numerical results show that the new scheme can achieve the optimal convergence order and maintain the nonnegativity of the solution for generalized polyhedral meshes and arbitrary diffusion tensors.

A Positivity-Preserving Finite Volume Scheme for Nonequilibrium Radiation Diffusion Equations on Distorted Meshes.

In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme.

Numerical Study of Near-Infrared Light Propagation in Aqueous Alumina Suspensions Using the Steady-State Radiative Transfer Equation and Dependent Scattering Theory

Understanding light propagation in liquid phantoms, such as colloidal suspensions, involves fundamental research of near-infrared optical imaging and spectroscopy for biological tissues. Our objective is to numerically investigate light propagation in the alumina colloidal suspensions with the mean alumina particle diameter of 55 nm at the volume fraction range 1–20%. We calculated the light scattering properties using the dependent scattering theory (DST) on a length scale comparable to the optical wavelength. We calculated the steady-state radiative transfer and photon diffusion equations (RTE and PDE) using the DST results based on the finite difference method in a length scale of the mean free path. The DST calculations showed that the scattering and reduced scattering coefficients become more prominent at a higher volume fraction. The anisotropy factor is almost zero at all the volume fractions, meaning the scattering is isotropic. The comparative study of the RTE with the PDE showed that the diffusion approximation holds at the internal region with all the volume fractions and the boundary region with the volume fraction higher than 1%. Our findings suggest the usefulness of the PDE as a light propagation model for the alumina suspensions rather than the RTE, which provides accurate but complicated computation.

An extremum‐preserving finite volume scheme for three‐temperature radiation diffusion equations

In this paper, an extremum‐preserving finite volume scheme is constructed for the two‐dimensional three‐temperature (2D 3‐T) radiation diffusion equations. The harmonic averaging points located at cell edge are applied to define the auxiliary unknowns, and the primary unknowns are defined at cell center. This scheme has a fixed stencil and satisfies the local conservation condition and discrete extremum principle. The existence of discrete solution is proved by using the fixed point theorem. Moreover, the stability analysis of this scheme is also presented. Numerical results illustrate that this scheme is efficient and accurate in solving the 2D 3‐T radiation diffusion equations on distorted meshes.

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

Two-Group Radiative Transfer Benchmarks for the Non-Equilibrium Diffusion Model

Semi-analytic solutions for high-energy density radiation diffusion problems in slab geometry using a two-group model for the frequency (photon energy) variable are presented. To obtain these solutions we specify forms for the heat capacity and emissivity in the high energy group that are a function of the fraction of radiation emission in the low energy group in order to linearize the problem. This results in a linear system of equations that are solved via Laplace and Fourier transforms: the Laplace transform is inverted analytically and the inverse Fourier transform is computed using numerical integration. It is demonstrated that these solutions can be useful in verifying codes for solving the radiation diffusion equations in the high-energy density regime. Additionally, we include solutions for an optically thick problem that can be used to test the asymptotic diffusion limit of codes solving a transport model for radiative transfer.

Multi-core parallel BRSMF method for 2D3T radiation diffusion equations

Multi-core parallel BRSMF method for 2D3T radiation diffusion equations

Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.