Abstract
A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.
Highlights
Accurate numerical solution of the steady-state monoenergetic particle transport equation in the deterministic scope is one of the key research topics in the nuclear computational science community
We focus on the angular discretization error caused by the widely-applied Discrete Ordinates (SN ) method in the numerical solution of the transport equation
Based on the regularity of the exact angular flux and the conventional Simpson’s rule, we developed a new quadrature type that can treat the interior singularities inside the integration interval to achieve faster error convergence rate than the traditional angular quadrature types
Summary
Accurate numerical solution of the steady-state monoenergetic particle transport equation in the deterministic scope is one of the key research topics in the nuclear computational science community. Both spatial and angular variables need to be discretized, and the applied discretization methods introduce discretization errors into the numerical solution. Motivated by the conclusion from that work, we realized that it is the regularity of the exact angular flux that governs the angular quadrature error convergence rate. Based on the regularity of the exact angular flux and the conventional Simpson’s rule, we developed a new quadrature type that can treat the interior singularities inside the integration interval to achieve faster error convergence rate than the traditional angular quadrature types
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