ANS is committed to advancing, fostering, and promoting the development and application of nuclear sciences and technologies to benefit society.
Explore the many uses for nuclear science and its impact on energy, the environment, healthcare, food, and more.
Explore membership for yourself or for your organization.
Conference Spotlight
Nuclear Energy Conference & Expo (NECX)
September 8–11, 2025
Atlanta, GA|Atlanta Marriott Marquis
Standards Program
The Standards Committee is responsible for the development and maintenance of voluntary consensus standards that address the design, analysis, and operation of components, systems, and facilities related to the application of nuclear science and technology. Find out What’s New, check out the Standards Store, or Get Involved today!
Latest Magazine Issues
Aug 2025
Jan 2025
Latest Journal Issues
Nuclear Science and Engineering
September 2025
Nuclear Technology
Fusion Science and Technology
August 2025
Latest News
Powering the future: How the DOE is fueling nuclear fuel cycle research and development
As global interest in nuclear energy surges, the United States must remain at the forefront of research and development to ensure national energy security, advance nuclear technologies, and promote international cooperation on safety and nonproliferation. A crucial step in achieving this is analyzing how funding and resources are allocated to better understand how to direct future research and development. The Department of Energy has spearheaded this effort by funding hundreds of research projects across the country through the Nuclear Energy University Program (NEUP). This initiative has empowered dozens of universities to collaborate toward a nuclear-friendly future.
K. D. Lathrop
Nuclear Science and Engineering | Volume 134 | Number 3 | March 2000 | Pages 239-264
Technical Paper | doi.org/10.13182/NSE00-A2114
Articles are hosted by Taylor and Francis Online.
To investigate errors caused by angular differencing in approximating the streaming terms of the transport equation, five different approximations are evaluated for three test problems in one-dimensional spherical geometry. The following schemes are compared: diamond, special truncation error minimizing weighted diamond, linear continuous (the original SN scheme), linear discontinuous, and new quadratic continuous. To isolate errors caused by angular differencing, the approximations are derived from the transport equation without spatial differencing, and the resulting coupled ordinary differential equations (ODEs) are solved with an ODE solver. Results from the approximations are compared with analytic solutions derived for two-region purely absorbing spheres. Most of the approximations are derived by taking moments of the conservation form of the transport equation. The quadratic continuous approximation is derived taking the zeroth moment of both the transport equation and the first angular derivative of the transport equation. The advantages of this approach are described. In all of the approximations, the desirability is shown of using an initializing computation of the = -1 angular flux to correctly compute the central flux and of having a difference approximation that ensures this central flux is the same for all directions. The behavior of the standard discrete ordinates equations in the diffusion limit is reviewed, and the linear and quadratic continuous approximations are shown to have the correct diffusion limit if an equal interval discrete quadrature is used.In all three test problems, the weighted diamond difference approximation has smaller maximum and average relative flux errors than the diamond or the linear continuous difference approximations. The quadratic continuous approximation and the linear discontinuous approximation are both more accurate than the other approximations, and the quadratic continuous approximation has a decided edge over the linear discontinuous approximation in relative flux errors. The diamond, weighted diamond, and linear continuous approximations show quadratic system absorption and system leakage error reduction behavior with increasing N. The linear discontinuous and quadratic continuous approximations show fourth-order error reduction in these quantities. In one of the two-region test problems, the slope of the exact angular flux changes from nearly vertical to nearly horizontal at those points in the exterior region at which the interior region source just becomes visible. At these spatial points, errors in the continuous approximations propagate to each successive outgoing direction, leading to an oscillatory spatial error. The discontinuous approximation does not propagate these errors, although errors near the point of rapid slope change are larger than in the quadratic continuous approximation.