A Nonlinear Positive Extension of the Linear Discontinuous Spatial Discretization of the Transport Equation
Maginot, Peter Gregory
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Linear discontinuous (LD) spatial discretization of the transport operator can generate negative angular flux solutions. In slab geometry, negativities are limited to optically thick cells. However, in multi-dimension problems, negativities can even occur in voids. Past attempts to eliminate the negativities associated with LD have focused on inherently positive solution shapes and ad-hoc fixups. We present a new, strictly non-negative finite element method that reduces to the LD method whenever the LD solution is everywhere positive. The new method assumes an angular flux distribution, e , that is a linear function in space, but with all negativities set-to- zero. Our new scheme always conserves the zeroth and linear spatial moments of the transport equation. For these reasons, we call our method the consistent set-to-zero (CSZ) scheme. CSZ can be thought of as a nonlinear modification of the LD scheme. When the LD solution is everywhere positive within a cell, psi csz = psi LD. If psi LD < 0 somewhere within a cell, psi csz is a linear function psi csz with all negativities set to zero. Applying CSZ to the transport moment equations creates a nonlinear system of equations which is solved to obtain a non-negative solution that preserves the moments of the transport equation. These properties make CSZ unique; it encompasses the desirable properties of both strictly positive nonlinear solution representations and ad-hoc fixups. Our test problems indicate that CSZ avoids the slow spatial convergence properties of past inherently positive solutions representations, is more accurate than ad-hoc fixups, and does not require significantly more computational work to solve a problem than using an ad-hoc fixup. Overall, CSZ is easy to implement and a valuable addition to existing transport codes, particularly for shielding applications. CSZ is presented here in slab and rect- angular geometries, but is readily extensible to three-dimensional Cartesian (brick) geometries. To be applicable to other simulations, particularly radiative transfer, additional research will need to be conducted, focusing on the diffusion limit in multi-dimension geometries and solution acceleration techniques.