Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

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Abstract

This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.

LanguageEnglish
Pages219-239
Number of pages21
JournalComputer Methods in Applied Mechanics and Engineering
Volume325
DOIs
StatePublished - 1 Oct 2017

Fingerprint

Boltzmann equation
Error analysis
hierarchies
moments
Adaptive algorithms
approximation
Heat flux
Heat transfer
error analysis
estimates
functionals
heat flux
heat transfer
shock

Keywords

  • A-posteriori error estimation
  • Boltzmann equation
  • Discontinuous Galerkin finite element methods
  • Goal-oriented model adaptivity
  • Hyperbolic systems
  • Moment systems

Cite this

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title = "Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation",
abstract = "This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.",
keywords = "A-posteriori error estimation, Boltzmann equation, Discontinuous Galerkin finite element methods, Goal-oriented model adaptivity, Hyperbolic systems, Moment systems",
author = "M.R.A. Abdelmalik and {van Brummelen}, E.H.",
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AB - This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.

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