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

TL;DR

This paper develops an a-posteriori error analysis and adaptive algorithm for goal-oriented approximations of the steady Boltzmann equation, combining moment-system and discontinuous Galerkin methods to optimize specific functionals.

Contribution

It introduces a novel goal-oriented error estimation framework and adaptive refinement strategy for Boltzmann equation approximations using moment and DGFE methods.

Findings

Effective local refinement of moment model order improves heat flux approximation.

Error bounds guide adaptive algorithms for goal-specific accuracy.

Validated on heat transfer and shock structure problems.

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.