Adaptive discrete-ordinates algorithms and strategies

Abstract

The approaches for discretizing the direction variable in particle transport calculations are the discrete-ordinates method and function-expansion methods. Both approaches are limited if the transport solution is not smooth. Angular discretization errors in the discrete-ordinates method arise from the inability of a given quadrature set to accurately perform the needed integrals over the direction ("angular") domain. We propose that an adaptive discrete-ordinate algorithm will be useful in many problems of practical interest. We start with a "base quadrature set" and add quadrature points as needed in order to resolve the angular flux function. We compare an interpolated angular-flux value against a calculated value. If the values are within a user specified tolerance, the point is not added; otherwise it is. Upon the addition of a point we must recalculate weights. Our interpolatory functions map angular-flux values at the quadrature directions to a continuous function that can be evaluated at any direction. We force our quadrature weights to be consistent with these functions in the sense that the quadrature integral of the angular flux is the exact integral of the interpolatory function (a finite-element methodology that determines coefficients by collocation instead of the usual weightedresidual procedure). We demonstrate our approach in two-dimensional Cartesian geometry, focusing on the azimuthal direction The interpolative methods we test are simple linear, linear in sine and cosine, an Abu-Shumays “base” quadrature with a simple linear adaptive and an Abu-Shumays “base” quadrature with a linear in sine and cosine adaptive. In the latter two methods the local refinement does not reduce the ability of the base set to integrate high-order spherical harmonics (important in problems with highly anisotropic scattering). We utilize a variety of one-group test problems to demonstrate that in all cases, angular discretization errors (including "ray effects") can be eliminated to whatever tolerance the user requests. We further demonstrate through detailed quantitative analysis that local refinement does indeed produce a more efficient placement of unknowns. We conclude that this work introduces a very promising approach to a long-standing problem in deterministic transport, and we believe it will lead to fruitful avenues of further investigation.