Non-equilibrium Radiation Diffusion Research Articles

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.

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.

Asymptotic-Preserving Discrete Schemes for Non-Equilibrium Radiation Diffusion Problem in Spherical and Cylindrical Symmetrical Geometries

Asymptotic-Preserving Discrete Schemes for Non-Equilibrium Radiation Diffusion Problem in Spherical and Cylindrical Symmetrical Geometries

An efficient solving method for nonlinear convection diffusion equation

PurposeThis paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated.Design/methodology/approachA nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy.FindingsTheoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained.Practical implicationsThe numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems.Originality/valueThe FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

Asymptotic analysis of discrete schemes for non-equilibrium radiation diffusion

Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in 2 to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion. Provide AP fully discrete schemes for non-equilibrium radiation diffusion.Propose second order accurate schemes by asymmetric approach for boundary flux-limiter.Show first order AP property of spatially and fully discrete schemes with IB evolution.Devise subtle artificial solutions; verify accuracy and AP property quantitatively.Ideas can be generalized to 3-dimensional problems and higher order implicit schemes.

Monotone finite point method for non-equilibrium radiation diffusion equations

In this paper, we propose the monotone tailored-finite-point method for solving the non-equilibrium radiation diffusion equations. We first give two tailored finite point schemes for the nonlinear parabolic equation in one-dimensional case, then extend the idea to solve the radiation diffusion problem in 1D as well as 2D. By using variable substitute, our method satisfies the discrete maximum principle automatically, thus preserves the properties of monotonicity and positivity. Numerical results show that our method can capture the sharp front and can be accommodated to discontinues diffusion coefficient.

TRHD: Three-temperature radiation-hydrodynamics code with an implicit non-equilibrium radiation transport using a cell-centered monotonic finite volume scheme on unstructured-grids

Three-temperature (3T), unstructured-mesh, non-equilibrium radiation hydrodynamics (RHD) code have been developed for the simulation of intense thermal radiation or high-power laser driven radiative shock hydrodynamics in two-dimensional (2D) axis-symmetric geometries. The governing hydrodynamics equations are solved using a compatible unstructured Lagrangian method based on a control volume differencing (CVD) scheme. A second-order predictor–corrector (PC) integration scheme is used for the temporal discretization of the hydrodynamics equations. For the radiation energy transport, frequency averaged gray model is used in which the flux-limited diffusion (FLD) approximation is used to recover the free-streaming limit of the radiation propagation in optically thin regions. The proposed RHD model allows to have different temperatures for the electrons and ions. In addition to this, the electron and thermal radiation temperatures are assumed to be in non-equilibrium. Therefore, the thermal relaxation between the electrons and ions and the coupling between the radiation and matter energies are required to be computed self-consistently. For this, the coupled flux limited electron heat conduction and the non-equilibrium radiation diffusion equations are solved simultaneously by using an implicit, axis-symmetric, cell-centered, monotonic, nonlinear finite volume (NLFV) scheme. In this paper, we have described the details of the 2D, 3T, non-equilibrium RHD code developed along with a suite of validation test problems to demonstrate the accuracy and performance of the algorithms. We have also conducted a performance analysis with different linearity preserving interpolation schemes that are used for the evaluation of the nodal values in the NLFV scheme. Finally, in order to demonstrate full capability of the code implementation, we have presented the simulation of laser driven thin Aluminum (Al) foil acceleration. The simulation results are found to be in good agreement with the known solutions of RHD problems.

Dynamic implicit 3D adaptive mesh refinement for non-equilibrium radiation diffusion

The time dependent non-equilibrium radiation diffusion equations are important for solving the transport of energy through radiation in optically thick regimes and find applications in several fields including astrophysics and inertial confinement fusion. The associated initial boundary value problems that are encountered often exhibit a wide range of scales in space and time and are extremely challenging to solve. To efficiently and accurately simulate these systems we describe our research on combining techniques that will also find use more broadly for long term time integration of nonlinear multi-physics systems: implicit time integration for efficient long term time integration of stiff multi-physics systems, local control theory based step size control to minimize the required global number of time steps while controlling accuracy, dynamic 3D adaptive mesh refinement (AMR) to minimize memory and computational costs, Jacobian Free Newton–Krylov methods on AMR grids for efficient nonlinear solution, and optimal multilevel preconditioner components that provide level independent solver convergence.

The preconditioned Jacobian-free Newton-Krylov methods for nonequilibrium radiation diffusion equations

In general, it is difficult to use the Newton-Krylov methods to solve the large-scale multi-variable nonequilibrium reaction-diffusion systems. In this paper, by employing two new semi-implicit dis...

Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

Analytical benchmark for non-equilibrium radiation diffusion in finite size systems

Non-equilibrium radiation diffusion is an important mechanism of energy transport in inertial confinement fusion, astrophysical plasmas, furnaces and heat exchangers. In this paper, an analytical solution to the non-equilibrium Marshak diffusion problem in a planar slab and spherical shell of finite thickness is presented. Using Laplace transform method, the radiation and material energy densities are obtained as functions of space and time. The variation in integrated energy densities and leakage currents are also studied. In order to linearize the radiation transport and material energy equation, the heat capacity is assumed to be proportional to the cube of the material temperature and the opacity to be independent of temperature. The steady state energy densities show linear variation along the depth of the planar slab, whereas non-linear dependence is observed for the spherical shell. The analytical energy densities show good agreement with those obtained from finite difference method using small mesh width and time step. The benchmark results obtained in this work can be used to validate and verify non-equilibrium radiation diffusion computer codes in both planar and spherical geometry.

The preconditioned Jacobian-free Newton–Krylov methods for nonequilibrium radiation diffusion equations

In general, it is difficult to use the Newton–Krylov methods to solve the large-scale multi-variable nonequilibrium reaction–diffusion systems. In this paper, by employing two new semi-implicit discretization schemes to construct the preconditioners, the preconditioned Newton–Krylov methods are presented to solve the multidimensional problems. These methods cannot only improve the number of iterations, but also speed up the convergence of solutions. Numerical results are given to illustrate the effectiveness.

On linearization and preconditioning for radiation diffusion coupled to material thermal conduction equations

Jacobian-free Newton–Krylov (JFNK) method is an effective algorithm for solving large scale nonlinear equations. One of the most important advantages of JFNK method is that there is no necessity to form and store the Jacobian matrix of the nonlinear system when JFNK method is employed. However, an approximation of the Jacobian is needed for the purpose of preconditioning. In this paper, JFNK method is employed to solve a class of non-equilibrium radiation diffusion coupled to material thermal conduction equations, and two preconditioners are designed by linearizing the equations in two methods. Numerical results show that the two preconditioning methods can improve the convergence behavior and efficiency of JFNK method.

Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem

AbstractFor a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

Monotone Finite Volume Schemes of Nonequilibrium Radiation Diffusion Equations on Distorted Meshes

We construct a monotone finite volume scheme on distorted meshes for multimaterial, nonequilibrium radiation diffusion problems, which are described by the coupled radiation diffusion and material conduction equations. Moreover, we prove theoretically that the scheme is monotone. Numerical results are presented to show that our scheme preserves positivity of solution on various distorted meshes, and the contours of numerical solution obtained by our scheme on distorted meshes accord with that on rectangular meshes. Moreover, numerical tests indicate that our monotone scheme is more computationally efficient than the nine point scheme. These results show that our nonlinear monotone finite volume scheme is a practical and attractive method for solving nonlinear diffusion equations on distorted meshes.

Numerical analysis of time integration errors for nonequilibrium radiation diffusion

Numerical analysis of time integration errors for nonequilibrium radiation diffusion is considered. Two first-order implicit time integration methods are studied. Asymptotic analysis and modified equation analysis are applied to both time integration methods. Numerical experiments are used to highlight the results of the analysis. Asymptotic analysis is used to highlight the source of temperature spiking when a hot radiation wave propagates into a cold material. Modified equation analysis is used to provide insight into the thermal wave speed coming from the two different first-order methods.

Efficient solution of multi-dimensional flux-limited nonequilibrium radiation diffusion coupled to material conduction with second-order time discretization

Many algorithms for the second-order time evolution of the coupled radiation diffusion and material conduction equations have been published. Most of them are cumbersome to implement and much slower computationally than their first-order equivalent algorithms. This paper presents a simpler approach that is both computationally efficient and easy to implement. Second-order behavior can be achieved even when the iteration at each time step is incompletely converged. The test problem uses multiple materials and nonlinear heat capacities. Unexpectedly, details in the discretization of the gradient in the flux limiter significantly affect the spatial and temporal convergence of the solution.

Temporal Accuracy of the Nonequilibrium Radiation Diffusion Equations Applied to Two-Dimensional Multimaterial Simulations

A study of the temporal accuracy of a variety of first- and second-order time-integration methods applied to two-dimensional, multimaterial, nonequilibrium, radiation diffusion simulations is presented. These methods are categorized by their temporal order of accuracy, whether the algorithm includes operator splitting, and whether the algorithm includes linearizations. Results are presented that simultaneously measure accuracy and efficiency of the different methods on two different test problems. The two test problems are designed to represent an easy problem, where different approximations may be accurate, and a hard test problem that will stress the different solution algorithms. Results show the importance of being second-order accurate in time and the importance of time-step control.

A multigrid preconditioner and automatic differentiation for non-equilibrium radiation diffusion problems

We study the efficient solution of non-equilibrium radiation diffusion problems. An implicit time discretization leads to the solution of systems of non-linear equations which couple radiation energy and material temperature. We consider the implicit Euler method, the mid-point scheme, the two-step backward differentiation formula, and a two-stage implicit Runge–Kutta method for time discretization. We employ a Newton–Krylov method in the solution of arising non-linear problems. We describe the computation of the Jacobian matrix for Newton’s method using automatic differentiation based on the operator overloading in Fortran 90. For GMRES iterations, we propose a simple multigrid preconditioner applied directly to the coupled linearized problems. We demonstrate the efficiency and scalability of the proposed solution procedure by solving one-dimensional and two-dimensional model problems.

Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration

We present a solution method for fully implicit radiation diffusion problems discretized on meshes having millions of spatial zones. This solution method makes use of high order in time integration techniques, inexact Newton–Krylov nonlinear solvers, and multigrid preconditioners. We explore the advantages and disadvantages of high order time integration methods for the fully implicit formulation on both two- and three-dimensional problems with tabulated opacities and highly nonlinear fusion source terms.