Nonnegative Methods for Bilinear Discontinuous Differencing of the S_N Equations on Quadrilaterals

Abstract

Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional S_N equations. Though matrix lumping inhibits negative angular flux solutions of the S_N equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions.

We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. The fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both fixups.

• Radiation transport
• discontinuous finite element method
• nonnegative

Acknowledgments

The authors wish to thank W. D. Hawkins for his invaluable and timely assistance in implementing and testing these methods. At nonoverlapping times, the work of P. G. Maginot was funded by the U.S. Department of Energy (DOE) Computational Science Graduate Fellowship program, administered by the Krell Institute, under grant DE-FG02-97ER25308, or conducted under the auspices of DOE by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. The work of J. C. Ragusa and J. E. Morel was supported in part by the Center for Exascale Radiation Transport, under DOE, National Nuclear Security Administration, award number DE-NA0002376.