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

•A variable order spherical harmonics scheme is presented.•An adaptive process is proposed to automatically refine the angular resolution.•A regular error estimator and a goal-based error estimator are presented.•The adaptive methods are applied to fixed source and eigenvalue problems.•Adaptive methods give more accurate solutions than uniform angular resolution. 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.