Ray effect mitigation for the discrete ordinates method through quadrature rotation

Abstract

Solving the radiation transport equation is a challenging task, due to the high dimensionality of the solution's phase space. The commonly used discrete ordinates (S N ) method suffers from ray effects which result from a break in rotational symmetry from the finite set of directions chosen by S N . The spherical harmonics (P N ) equations, on the other hand, preserve rotational symmetry, but can produce negative particle densities. The discrete ordinates (S N ) method, in turn, by construction ensures non-negative particle densities.In this paper we present a modified version of the S N method, the rotated S N (rS N ) method. Compared to S N , we add a rotation and interpolation step for the angular quadrature points and the respective function values after every time step. Thereby, the number of directions on which the solution evolves is effectively increased and ray effects are mitigated. Solution values on rotated ordinates are computed by an interpolation step. Implementation details are provided and in our experiments the rotation and interpolation step only adds 5% to 10% to the runtime of the S N method. We apply the rS N method to the line-source and a lattice test case, both being prone to ray effects. Ray effects are reduced significantly, even for small numbers of quadrature points. The rS N method yields qualitatively similar solutions to the S N method with less than a third of the number of quadrature points, both for the line-source and the lattice problem. The code used to produce our results is freely available and can be downloaded [4].

Ray Effect in Rarefied Flow Simulation

“…Similar discussions can be found in radiative transfer in the discretization of angular variables [15]. In addition, the mitigation methods which are equivalent to obtaining the solutions with more discrete velocities can be employed as well, such as taking averaged solutions over multiple times of computations on different sets of velocity space discretization [11][12][13].…”

“…However, for the standard quadrature sets of DOM, undesirable negative weights will occur for N > 12, which gives an upper limit on the number of angular directions and constrains the S N method to fully remove the ray effect. Therefore, many strategies have been proposed to adjust the angular discretization and quadrature sets in order to mitigate the ray effect, such as T N quadrature set [10], staggered and adaptive quadrature sets [11], and quadrature rotation [12,13]. Detailed analysis and reviews can be found in [14][15][16].…”

Unified gas-kinetic wave-particle methods III: Multiscale photon transport

“…This is a particularly difficult problem for the S n based methods because they suffer from severe ray effect. Previous studies have used this problem for comparisons between methods [38] and as a test case for schemes to mitigate the ray effect [39]. Both the UGKWP method and the Monte Carlo method use 201 × 201 cells in space.…”

Ray Effects and False Scattering in Improved Discrete Ordinates Method

“…In terms of ray effects, in addition to increase discrete directions, to modify direction layout is another common remedy [11][12][13][14][15]. Numerical experiments have been conducted by Li et al [11] showing that ray effects can be reduced by modifying direction layouts, and three corresponding countermeasures were proposed in their paper.…”

“…Tencer [12] proposed a ray effects mitigation technique by averaging the computed results of various direction layouts, which are generated by arbitrarily rotating the reference layout. Quadrature rotation was also applied to solve time-dependent transport and results showed that ray effects were reduced significantly, including for small numbers of quadrature points [13]. Recently, goal-oriented angular adaptive algorithm was also utilized to mitigate ray effects [14,15].…”