R.P. Smedley-Stevenson

The Inner-Element Subgrid Scale Finite Element Method for the Boltzmann Transport Equation

Abstract This paper presents a new multiscale radiation transport method based on a Galerkin finite element spatial discretization of the Boltzmann transport equation. The approach incorporates a discontinuous subgrid scale (SGS) solution within the continuous finite element representation of the spatial variables. While the conventional discontinuous Galerkin (DG) method provides accurate and numerically stable solutions that suppress unphysical oscillations, the number of unknowns is relatively high. The key advantage of the proposed SGS approach is that the solutions are represented within the continuous finite element space, and therefore, the number of unknowns compared with DG is relatively low. The applications of this method are explored using linear finite elements, and some of the advantages of this new discretization over standard Petrov-Galerkin methods are demonstrated. The numerical examples are chosen to be demanding steady-state mono-energetic radiation transport problems that are likely to form unphysical oscillations within numerical scalar flux solutions. The numerical examples also provide evidence that the SGS method has a thick diffusion limit. This method is designed to work under arbitrary angular discretizations, so solutions using both spherical harmonics and discrete ordinates are presented.

Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation

Abstract A new method for applying anisotropic resolution in the angular domain of the Boltzmann transport equation is presented. The method builds on our previous work in which two spherical wavelet bases were developed for representing the direction of neutral particle travel. The method proposed here enables these wavelet bases to vary their angular approximations so that fine resolution may be applied only to the areas of the unit sphere (representing the direction of particle travel) that are important. We develop an error measure that operates in conjunction with the wavelet bases to determine this importance. A procedure by which the angular resolution is gradually refined for steady-state problems is also given. The adaptive wavelets are applied to three test problems that demonstrate the ability of the wavelets to resolve complex fluxes with relatively few functions, and to achieve this a particular emphasis is placed on their ability to approximate particle streaming through ducts with voids. It is shown that the wavelets are capable of applying the appropriate resolution (as dictated by the error measure) to the directional component of the angular flux at all spatial positions. This method therefore offers a new and highly efficient adaptive angular approximation method.

Finite Element Based Riemann Solvers for Time-Dependent and Steady-State Radiation Transport

Abstract A high-order, nonoscillatory scheme is described which solves the transient and steady-state Boltzmann transport equation on unstructured Finite Element (FE) meshes. Flux limiters are applied in the space and time domains resulting in a scheme which is both free from oscillations and globally high-order accurate in space and time. The method described is finite volume based and uses a consistent FE representation of the solution variables to obtain a high-order solution along the control volume boundaries. Careful inspection of the eigenstructure of the Riemann problem allows one to switch smoothly between a high-order and a low-order nonoscillatory solution.

Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation. Wavelet angular discretisation used to solve transport equation.Adaptive method developed for the wavelet discretisation.Anisotropic angular resolution demonstrated through the adaptive method.Adaptive method provides improvements in computational efficiency.

Asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport

Abstract This paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes, for a linear-discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns, in a process known as ‘source tilting’. The asymptotic limits of both Monte Carlo (continuous in space) and deterministic approaches (based on linear-discontinuous finite elements) for solving the transport equation are investigated in slab geometry. The resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell-edge discontinuity in the temperature representation. Based on this analysis it is possible to design accurate schemes for representing the material temperature, for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ALE (arbitrary Lagrange–Eulerian) hydrodynamics schemes.