Gordon L. Olson

A rapidly convergent iterative solution of the non-LTE line radiation transfer problem

Abstract An iterative scheme has been developed for the solution of the non-LTE line radiation transfer problem. The method uses an approximate operator that is deliberately chosen to be local so that it can be easily extended to multidimensional geometry. The difference between the formal and approximate solutions is used as a driving term for the iterations. In one-dimensional, semiinfinite and free-standing slabs, the technique is found to be very fast, robust, and applicable to a large class of problems.

Diffusion, P1, and other approximate forms of radiation transport

Abstract Full transport solutions of time-dependent problems can be computationally very expensive. Therefore, considerable effort has been devoted to developing approximate solution techniques that are much faster computationally and yet are accurate enough for a particular application. Many of these approximate solutions have been used in isolated problems and have not been compared to each other. This paper presents two test problems that test and compare several approximate transport techniques. In addition to the diffusion and P1 approximations, we will test several different flux-limited diffusion theories and variable Eddington factor closures. For completeness, we will show some variations that have not yet appeared in the literature that have some interesting consequences. For example, we have found a trivial way to modify the P1 equations to get the correct propagation velocity of a radiation front in the optically thin limit without modifying the accuracy of the solution in the optically thick limit. Also, we will demonstrate nonphysical behavior in some published techniques.

An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium

Abstract Benchmark solutions to nontrivial radiation transport problems are crucial to the validation of transport codes. This paper gives an analytical transport solution for non-equilibrium radiative transfer in an infinite and isotropically scattering medium. The radiation source in the medium is isotropic in angle and constant in time (but only exists in a finite period of time), and is allowed to be uniformly distributed in a finite space or to be located at a point. The solution is constructed by applying the Fourier transform with respect to spatial variable and the Laplace transform with respect to temporal variable. The integration over angular variable is treated exactly. The resulting solution, as a function of space and time and in the form of a double integral, is evaluated numerically without much difficulty. Tables and figures are given for the resulting benchmark solution.

An efficient nonlinear solution method for non-equilibrium radiation diffusion

Abstract A new nonlinear solution method is developed and applied to a non-equilibrium radiation diffusion problem. With this new method, Newton-like super-linear convergence is achieved in the nonlinear iteration, without the complexity of forming or inverting the Jacobian from a standard Newton method. The method is a unique combination of an outer Newton-based iteration and and inner conjugate gradient-like (Krylov) iteration. The effects of the Jacobian are probed only through approximate matrix–vector products required in the conjugate gradient-like iteration. The methodology behind the Jacobian-free Newton–Krylov method is given in detail. It is demonstrated that a simple, successive substitution, linearization produces an effective preconditioning matrix for the Krylov method. The efficiencies of different methods are compared and the benefits of converging the nonlinearities within a time step are demonstrated.

Nonlinear convergence, accuracy, and time step control in nonequilibrium radiation diffusion

Abstract We study the interaction between converging the nonlinearities within a time step and time step control, on the accuracy of nonequilibrium radiation diffusion calculations. Typically, this type of calculation is performed using operator-splitting where the nonlinearities are lagged one time step. This method of integrating the nonlinear system results in an “effective” time-step constraint to obtain accuracy. A time-step control that limits the change in dependent variables (usually energy) per time step is used. We investigate the possibility that converging the nonlinearities within a time step may allow significantly larger time-step sizes and improved accuracy as well. The previously described Jacobian-free Newton–Krylov method (JQSRT 63 (1999) 15) is used to converge all nonlinearities within a time step. In addition, a new time-step control method, based on the hyperbolic model of a thermal wave (J. Comput. Phys. 152 (1999) 790), is employed. The benefits and cost of a second-order accurate time step are considered. It is demonstrated that for a chosen accuracy, significant increases in solution efficiency can be obtained by converging nonlinearities within a time step.

Benchmark results for the non-equilibrium Marshak diffusion problem

Abstract As an extension of previous work in the literature, this paper considers a particular one-dimensional, halfspace, non-equilibrium Marshak wave problem. The radiative transfer model employed is a one-group diffusion approximation with Marshak boundary condition, where the radiation and material fields are out of equilibrium. An analytic solution for the distribution of radiative energy and material temperature as a function of space and time to this problem is given and tables of numerical results are generated. These benchmark results, together with the previously published results, are useful as a reference for validating time-dependent radiation diffusion computer codes. A comparison with a finite difference solution is presented which shows excellent agreement when a fine spatial mesh and small time steps are used.

Non-grey benchmark results for two temperature non-equilibrium radiative transfer

Abstract Benchmark solutions to time-dependent radiative transfer problems involving non-equilibrium coupling to the material temperature field are crucial for validating time-dependent radiation transport codes. Previous efforts on generating analytical solutions to non-equilibrium radiative transfer problems were all restricted to the one-group grey model. In this paper, a non-grey model, namely the picket-fence model, is considered for a two temperature non-equilibrium radiative transfer problem in an infinite medium. The analytical solutions, as functions of space and time, are constructed in the form of infinite integrals for both the diffusion description and transport description. These expressions are evaluated numerically and the benchmark results are generated. The asymptotic solutions for large and small times are also derived in terms of elementary functions and are compared with the exact results. Comparisons are given between the transport and diffusion solutions and between the grey and non-grey solutions.

Second-order time evolution of PN equations for radiation transport

Using polynomials to represent the angular variation of the radiation intensity is usually referred to as the PN or spherical harmonics method. For infinite order, the representation is an exact solution of the radiation transport solution. For finite N, in some physical situations there are oscillations in the solution that can make the radiation energy density be negative. For small N, the oscillations may be large enough to force the material temperature to numerically have non-physical negative values. The second-order time evolution algorithm presented here allows for more accurate solutions with larger time steps; however, it also can resolve the negativities that first-order time solutions smear out. Therefore, artificial scattering is studied to see how it can be used to decrease the oscillations in low-order solutions and prevent negativities. Small amounts of arbitrary, non-physical scattering can significantly improve the accuracy of the solution to test problems. Flux-limited diffusion solutions can also be improved by including artificial scattering. One- and two-dimensional test results are presented.

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.

Alternate closures for radiation transport using Legendre polynomials in 1D and spherical harmonics in 2D

Highlights? The traditional closure causes waves to propagate slower than the speed of light. ? A new angle closure is presented that propagates waves exactly at c. ? Scaling the nth equation may greatly improve the accuracy of transport solutions. ? More accurate solutions are found without added computational cost. ? The same scaling procedure and scale factors work in one- and multi-dimensions. When using polynomial expansions for the angular variables in the radiation transport equation, the usual procedure is to truncate the series by setting all higher order terms to zero. At low order, such simple closures may not give the optimum solution. This work tests alternate closures that scale either the time- or spatial-derivatives in the highest order equation. These scale factors can be chosen such that waves propagate at exactly the speed of light in optically thin media. Alternatively, they may be chosen to significantly improve the accuracy of low-order solutions with no additional computational cost. The same scaling procedure and scale factors work in one- and multi-dimensions. In multidimensions, reducing the order of a solution can save significant amounts of computer time.