Goal-based angular adaptivity applied to the spherical harmonics discretisation of the neutral particle transport equation

Abstract

A variable order spherical harmonics scheme has been described and employed for the solution of the neutral particle transport equation. The scheme is specifically described with application within the inner-element sub-grid scale finite element spatial discretisation. The angular resolution is variable across both the spatial and energy dimensions. That is, the order of the spherical harmonic expansion may differ at each node of the mesh for each energy group. The variable order scheme has been used to develop adaptive methods for the angular resolution of the particle transport phase-space. Two types of adaptive method have been developed and applied to examples. The first is regular adaptivity, in which the error in the solution over the entire domain is minimised. The second is goal-based adaptivity, in which the error in a specified functional is minimised. The methods were applied to fixed source and eigenvalue examples. Both methods demonstrate an improved accuracy for a given number of degrees of freedom in the angular discretisation.

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

“…. (35) where h [q] (φ + [q] (r), φ − [q] (r)) is the numerical flux and +, − refer to the limit of the flux components inside and outside of the surface of the spatial element, respectively.…”

“…discretization. Their spatial discretization was combined with adaptive P n angular discretization as well [35]. The authors show various results for challenging transport problems where local refinement on the sphere leads to a reduction of the number of unknowns required for an accurate solution.…”

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

“…There has been less research into adaptive methods applied to the angular dimensions of the transport equation. Adaptivity applied to the P N method has been investigated in various forms by Park et al, Rupp et al and Goffin et al [20,23,11]. In addition, research into adaptive methods for the S N method has been carried out separately by Stone and Jarrell [25,14].…”

Directions in Radiation Transport Modelling

“…This technology has the added advantage that the user need only generate a simple mesh that conforms to the problem geometry and the code will then adjust the mesh to an optimal one for the user's requirements. The technique has been extended from spatial mesh adaptivity to angular adaptivity [55]; in this context, the order of the angular representation can vary spatially to provide more efficient calculation than is possible with a fixed angular representation.…”