Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

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The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows ... continued below

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Densmore, Jeffery D; Warsa, James S; Lowrie, Robert B & Morel, Jim E January 1, 2008.

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The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

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  • Journal Name: Journal of Computational Physics; Journal Volume: 228; Journal Issue: 16

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  • Report No.: LA-UR-08-08070
  • Report No.: LA-UR-08-8070
  • Grant Number: AC52-06NA25396
  • Office of Scientific & Technical Information Report Number: 956682
  • Archival Resource Key: ark:/67531/metadc929624

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  • January 1, 2008

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  • Nov. 13, 2016, 7:26 p.m.

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  • Dec. 12, 2016, 1:31 p.m.

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Densmore, Jeffery D; Warsa, James S; Lowrie, Robert B & Morel, Jim E. Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation, article, January 1, 2008; [New Mexico]. (digital.library.unt.edu/ark:/67531/metadc929624/: accessed December 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.