A Linear, Exponential-Discontinuous Scheme for Discrete-Ordinates Calculations in Slab Geometry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929275882438656 |
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| author | Roberts, Jeremy A. |
| author_facet | Roberts, Jeremy A. |
| contents | Presented here is a preliminary study of a strictly linear, discontinuous-Petrov-Galerkin scheme for the discrete-ordinates method in slab geometry. By ``linear'', we mean the discretization does not depend on the solution itself as is the case in classical ``fix-up'' schemes and other nonlinear schemes that have been explored to maintain positive solutions with improved accuracy. By discontinuous, we mean the angular flux $ψ$ and scalar flux $ϕ$ are piecewise continuous functions that may exhibit discontinuities at cell boundaries. Finally, by ``Petrov-Galerkin,'' we mean a finite-element scheme in which the ``trial'' and ``test'' functions differ. In particular, we find that a trial basis consisting of a constant and exponential function that exactly represents the step-characteristic solution with a constant and linear test basis produces a scheme (1) with slightly better local errors than the linear-discontinuous (LD) scheme (for thin cells), (2) accuracy that approaches the linear-characteristic (LC) scheme (when the LC solution is positive), and (3) is positive as long as the first two source Legendre moments satisfy $|s_1| < 3 s_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_08816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Linear, Exponential-Discontinuous Scheme for Discrete-Ordinates Calculations in Slab Geometry Roberts, Jeremy A. Numerical Analysis Presented here is a preliminary study of a strictly linear, discontinuous-Petrov-Galerkin scheme for the discrete-ordinates method in slab geometry. By ``linear'', we mean the discretization does not depend on the solution itself as is the case in classical ``fix-up'' schemes and other nonlinear schemes that have been explored to maintain positive solutions with improved accuracy. By discontinuous, we mean the angular flux $ψ$ and scalar flux $ϕ$ are piecewise continuous functions that may exhibit discontinuities at cell boundaries. Finally, by ``Petrov-Galerkin,'' we mean a finite-element scheme in which the ``trial'' and ``test'' functions differ. In particular, we find that a trial basis consisting of a constant and exponential function that exactly represents the step-characteristic solution with a constant and linear test basis produces a scheme (1) with slightly better local errors than the linear-discontinuous (LD) scheme (for thin cells), (2) accuracy that approaches the linear-characteristic (LC) scheme (when the LC solution is positive), and (3) is positive as long as the first two source Legendre moments satisfy $|s_1| < 3 s_0$. |
| title | A Linear, Exponential-Discontinuous Scheme for Discrete-Ordinates Calculations in Slab Geometry |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.08816 |